\input texinfo    @c -*-texinfo-*-
@c %**start of header
@setfilename gmp.info
@documentencoding ISO-8859-1
@include version.texi
@settitle GNU MP @value{VERSION}
@synindex tp fn
@iftex
@afourpaper
@end iftex
@comment %**end of header

@copying
This manual describes how to install and use the GNU multiple precision
arithmetic library, version @value{VERSION}.

Copyright 1991, 1993-2016, 2018-2020 Free Software Foundation, Inc.

Permission is granted to copy, distribute and/or modify this document under
the terms of the GNU Free Documentation License, Version 1.3 or any later
version published by the Free Software Foundation; with no Invariant Sections,
with the Front-Cover Texts being ``A GNU Manual'', and with the Back-Cover
Texts being ``You have freedom to copy and modify this GNU Manual, like GNU
software''.  A copy of the license is included in
@ref{GNU Free Documentation License}.
@end copying
@c  Note the @ref above must be on one line, a line break in an @ref within
@c  @copying will bomb in recent texinfo.tex (eg. 2004-04-07.08 which comes
@c  with texinfo 4.7), with messages about missing @endcsname.


@c  Texinfo version 4.2 or up will be needed to process this file.
@c
@c  The version number and edition number are taken from version.texi provided
@c  by automake (note that it's regenerated only if you configure with
@c  --enable-maintainer-mode).
@c
@c  Notes discussing the present version number of GMP in relation to previous
@c  ones (for instance in the "Compatibility" section) must be updated at
@c  manually though.
@c
@c  @cindex entries have been made for function categories and programming
@c  topics.  The "mpn" section is not included in this, because a beginner
@c  looking for "GCD" or something is only going to be confused by pointers to
@c  low level routines.
@c
@c  @cindex entries are present for processors and systems when there's
@c  particular notes concerning them, but not just for everything GMP
@c  supports.
@c
@c  Index entries for files use @code rather than @file, @samp or @option,
@c  since the latter come out with quotes in TeX, which are nice in the text
@c  but don't look so good in index columns.
@c
@c  Tex:
@c
@c  A suitable texinfo.tex is supplied, a newer one should work equally well.
@c
@c  HTML:
@c
@c  Nothing special is done for links to external manuals, they just come out
@c  in the usual makeinfo style, eg. "../libc/Locales.html".  If you have
@c  local copies of such manuals then this is a good thing, if not then you
@c  may want to search-and-replace to some online source.
@c

@dircategory GNU libraries
@direntry
* gmp: (gmp).                   GNU Multiple Precision Arithmetic Library.
@end direntry

@c  html <meta name="description" content="...">
@documentdescription
How to install and use the GNU multiple precision arithmetic library, version @value{VERSION}.
@end documentdescription

@c smallbook
@finalout
@setchapternewpage on

@ifnottex
@node Top, Copying, (dir), (dir)
@top GNU MP
@end ifnottex

@iftex
@titlepage
@title GNU MP
@subtitle The GNU Multiple Precision Arithmetic Library
@subtitle Edition @value{EDITION}
@subtitle @value{UPDATED}

@author by Torbj@"orn Granlund and the GMP development team
@c @email{tg@@gmplib.org}

@c Include the Distribution inside the titlepage so
@c that headings are turned off.

@tex
\global\parindent=0pt
\global\parskip=8pt
\global\baselineskip=13pt
@end tex

@page
@vskip 0pt plus 1filll
@end iftex

@insertcopying
@ifnottex
@sp 1
@end ifnottex

@iftex
@end titlepage
@headings double
@end iftex

@c  Don't bother with contents for html, the menus seem adequate.
@ifnothtml
@contents
@end ifnothtml

@menu
* Copying::                    GMP Copying Conditions (LGPL).
* Introduction to GMP::        Brief introduction to GNU MP.
* Installing GMP::             How to configure and compile the GMP library.
* GMP Basics::                 What every GMP user should know.
* Reporting Bugs::             How to usefully report bugs.
* Integer Functions::          Functions for arithmetic on signed integers.
* Rational Number Functions::  Functions for arithmetic on rational numbers.
* Floating-point Functions::   Functions for arithmetic on floats.
* Low-level Functions::        Fast functions for natural numbers.
* Random Number Functions::    Functions for generating random numbers.
* Formatted Output::           @code{printf} style output.
* Formatted Input::            @code{scanf} style input.
* C++ Class Interface::        Class wrappers around GMP types.
* Custom Allocation::          How to customize the internal allocation.
* Language Bindings::          Using GMP from other languages.
* Algorithms::                 What happens behind the scenes.
* Internals::                  How values are represented behind the scenes.

* Contributors::               Who brings you this library?
* References::                 Some useful papers and books to read.
* GNU Free Documentation License::
* Concept Index::
* Function Index::
@end menu


@c  @m{T,N} is $T$ in tex or @math{N} otherwise.  Commas in N or T don't work,
@c  but @C{} can be used instead.
@iftex
@macro m {T,N}
@tex$\T\$@end tex
@end macro
@end iftex
@ifnottex
@macro m {T,N}
@math{\N\}
@end macro
@end ifnottex

@c  @mm{T,N} is $T$ tex and html and @math{N} in info.  Commas in N or T don't
@c  work, but @C{} can be used instead.
@iftex
@macro mm {T,N}
@tex$\T\$@end tex
@end macro
@end iftex

@ifhtml
@macro mm {T,N}
@math{\T\}
@end macro
@end ifhtml

@ifinfo
@macro mm {T,N}
@math{\N\}
@end macro
@end ifinfo


@macro C {}
,
@end macro

@c  @ms{V,N} is $V_N$ in tex or just vn otherwise.  This suits simple
@c  subscripts like @ms{x,0}.
@iftex
@macro ms {V,N}
@tex$\V\_{\N\}$@end tex
@end macro
@end iftex
@ifnottex
@macro ms {V,N}
\V\\N\
@end macro
@end ifnottex

@c  @nicode{S} is plain S in info, or @code{S} elsewhere.  This can be used
@c  when the quotes that @code{} gives in info aren't wanted, but the
@c  fontification in tex or html is wanted.  Doesn't work as @nicode{'\\0'}
@c  though (gives two backslashes in tex).
@ifinfo
@macro nicode {S}
\S\
@end macro
@end ifinfo
@ifnotinfo
@macro nicode {S}
@code{\S\}
@end macro
@end ifnotinfo

@c  @nisamp{S} is plain S in info, or @samp{S} elsewhere.  This can be used
@c  when the quotes that @samp{} gives in info aren't wanted, but the
@c  fontification in tex or html is wanted.
@ifinfo
@macro nisamp {S}
\S\
@end macro
@end ifinfo
@ifnotinfo
@macro nisamp {S}
@samp{\S\}
@end macro
@end ifnotinfo

@c  Usage: @GMPtimes{}
@c  Give either \times or the word "times".
@tex
\gdef\GMPtimes{\times}
@end tex
@ifnottex
@macro GMPtimes
times
@end macro
@end ifnottex

@c  Usage: @GMPmultiply{}
@c  Give * in info, or nothing in tex.
@tex
\gdef\GMPmultiply{}
@end tex
@ifnottex
@macro GMPmultiply
*
@end macro
@end ifnottex

@c  Usage: @GMPabs{x}
@c  Give either |x| in tex, or abs(x) in info or html.
@tex
\gdef\GMPabs#1{|#1|}
@end tex
@ifnottex
@macro GMPabs {X}
@abs{}(\X\)
@end macro
@end ifnottex

@c  Usage: @GMPfloor{x}
@c  Give either \lfloor x\rfloor in tex, or floor(x) in info or html.
@tex
\gdef\GMPfloor#1{\lfloor #1\rfloor}
@end tex
@ifnottex
@macro GMPfloor {X}
floor(\X\)
@end macro
@end ifnottex

@c  Usage: @GMPceil{x}
@c  Give either \lceil x\rceil in tex, or ceil(x) in info or html.
@tex
\gdef\GMPceil#1{\lceil #1 \rceil}
@end tex
@ifnottex
@macro GMPceil {X}
ceil(\X\)
@end macro
@end ifnottex

@c  Math operators already available in tex, made available in info too.
@c  For example @bmod{} can be used in both tex and info.
@ifnottex
@macro bmod
mod
@end macro
@macro gcd
gcd
@end macro
@macro ge
>=
@end macro
@macro le
<=
@end macro
@macro log
log
@end macro
@macro min
min
@end macro
@macro leftarrow
<-
@end macro
@macro rightarrow
->
@end macro
@end ifnottex

@c  New math operators.
@c  @abs{} can be used in both tex and info, or just \abs in tex.
@tex
\gdef\abs{\mathop{\rm abs}}
@end tex
@ifnottex
@macro abs
abs
@end macro
@end ifnottex

@c  @cross{} is a \times symbol in tex, or an "x" in info.  In tex it works
@c  inside or outside $ $.
@tex
\gdef\cross{\ifmmode\times\else$\times$\fi}
@end tex
@ifnottex
@macro cross
x
@end macro
@end ifnottex

@c  @times{} made available as a "*" in info and html (already works in tex).
@ifnottex
@macro times
*
@end macro
@end ifnottex

@c  Usage: @W{text}
@c  Like @w{} but working in math mode too.
@tex
\gdef\W#1{\ifmmode{#1}\else\w{#1}\fi}
@end tex
@ifnottex
@macro W {S}
@w{\S\}
@end macro
@end ifnottex

@c  Usage: \GMPdisplay{text}
@c  Put the given text in an @display style indent, but without turning off
@c  paragraph reflow etc.
@tex
\gdef\GMPdisplay#1{%
\noindent
\advance\leftskip by \lispnarrowing
#1\par}
@end tex

@c  Usage: \GMPhat
@c  A new \hat that will work in math mode, unlike the texinfo redefined
@c  version.
@tex
\gdef\GMPhat{\mathaccent"705E}
@end tex

@c  Usage: \GMPraise{text}
@c  For use in a $ $ math expression as an alternative to "^".  This is good
@c  for @code{} in an exponent, since there seems to be no superscript font
@c  for that.
@tex
\gdef\GMPraise#1{\mskip0.5\thinmuskip\hbox{\raise0.8ex\hbox{#1}}}
@end tex

@c  Usage: @texlinebreak{}
@c  A line break as per @*, but only in tex.
@iftex
@macro texlinebreak
@*
@end macro
@end iftex
@ifnottex
@macro texlinebreak
@end macro
@end ifnottex

@c  Usage: @maybepagebreak
@c  Allow tex to insert a page break, if it feels the urge.
@c  Normally blocks of @deftypefun/funx are kept together, which can lead to
@c  some poor page break positioning if it's a big block, like the sets of
@c  division functions etc.
@tex
\gdef\maybepagebreak{\penalty0}
@end tex
@ifnottex
@macro maybepagebreak
@end macro
@end ifnottex

@c  Usage: @GMPreftop{info,title}
@c  Usage: @GMPpxreftop{info,title}
@c
@c  Like @ref{} and @pxref{}, but designed for a reference to the top of a
@c  document, not a particular section.  The TeX output for plain @ref insists
@c  on printing a particular section, GMPreftop gives just the title.
@c
@c  The texinfo manual recommends putting a likely section name in references
@c  like this, eg. "Introduction", but it seems better to just give the title.
@c
@iftex
@macro GMPreftop{info,title}
@i{\title\}
@end macro
@macro GMPpxreftop{info,title}
see @i{\title\}
@end macro
@end iftex
@c
@ifnottex
@macro GMPreftop{info,title}
@ref{Top,\title\,\title\,\info\,\title\}
@end macro
@macro GMPpxreftop{info,title}
@pxref{Top,\title\,\title\,\info\,\title\}
@end macro
@end ifnottex


@node Copying, Introduction to GMP, Top, Top
@comment  node-name, next, previous,  up
@unnumbered GNU MP Copying Conditions
@cindex Copying conditions
@cindex Conditions for copying GNU MP
@cindex License conditions

This library is @dfn{free}; this means that everyone is free to use it and
free to redistribute it on a free basis.  The library is not in the public
domain; it is copyrighted and there are restrictions on its distribution, but
these restrictions are designed to permit everything that a good cooperating
citizen would want to do.  What is not allowed is to try to prevent others
from further sharing any version of this library that they might get from
you.@refill

Specifically, we want to make sure that you have the right to give away copies
of the library, that you receive source code or else can get it if you want
it, that you can change this library or use pieces of it in new free programs,
and that you know you can do these things.@refill

To make sure that everyone has such rights, we have to forbid you to deprive
anyone else of these rights.  For example, if you distribute copies of the GNU
MP library, you must give the recipients all the rights that you have.  You
must make sure that they, too, receive or can get the source code.  And you
must tell them their rights.@refill

Also, for our own protection, we must make certain that everyone finds out
that there is no warranty for the GNU MP library.  If it is modified by
someone else and passed on, we want their recipients to know that what they
have is not what we distributed, so that any problems introduced by others
will not reflect on our reputation.@refill

More precisely, the GNU MP library is dual licensed, under the conditions of
the GNU Lesser General Public License version 3 (see
@file{COPYING.LESSERv3}), or the GNU General Public License version 2 (see
@file{COPYINGv2}). This is the recipient's choice, and the recipient also has
the additional option of applying later versions of these licenses. (The
reason for this dual licensing is to make it possible to use the library with
programs which are licensed under GPL version 2, but which for historical or
other reasons do not allow use under later versions of the GPL).

Programs which are not part of the library itself, such as demonstration
programs and the GMP testsuite, are licensed under the terms of the GNU
General Public License version 3 (see @file{COPYINGv3}), or any later
version.


@node Introduction to GMP, Installing GMP, Copying, Top
@comment  node-name,  next,  previous,  up
@chapter Introduction to GNU MP
@cindex Introduction

GNU MP is a portable library written in C for arbitrary precision arithmetic
on integers, rational numbers, and floating-point numbers.  It aims to provide
the fastest possible arithmetic for all applications that need higher
precision than is directly supported by the basic C types.

Many applications use just a few hundred bits of precision; but some
applications may need thousands or even millions of bits.  GMP is designed to
give good performance for both, by choosing algorithms based on the sizes of
the operands, and by carefully keeping the overhead at a minimum.

The speed of GMP is achieved by using fullwords as the basic arithmetic type,
by using sophisticated algorithms, by including carefully optimized assembly
code for the most common inner loops for many different CPUs, and by a general
emphasis on speed (as opposed to simplicity or elegance).

There is assembly code for these CPUs:
@cindex CPU types
ARM Cortex-A9, Cortex-A15, and generic ARM,
DEC Alpha 21064, 21164, and 21264,
AMD K8 and K10 (sold under many brands, e.g. Athlon64, Phenom, Opteron)
Bulldozer, and Bobcat,
Intel Pentium, Pentium Pro/II/III, Pentium 4, Core2, Nehalem, Sandy bridge, Haswell, generic x86,
Intel IA-64,
Motorola/IBM PowerPC 32 and 64 such as POWER970, POWER5, POWER6, and POWER7,
MIPS 32-bit and 64-bit,
SPARC 32-bit ad 64-bit with special support for all UltraSPARC models.
There is also assembly code for many obsolete CPUs.


@cindex Home page
@cindex Web page
@noindent
For up-to-date information on GMP, please see the GMP web pages at

@display
@uref{https://gmplib.org/}
@end display

@cindex Latest version of GMP
@cindex Anonymous FTP of latest version
@cindex FTP of latest version
@noindent
The latest version of the library is available at

@display
@uref{https://ftp.gnu.org/gnu/gmp/}
@end display

Many sites around the world mirror @samp{ftp.gnu.org}, please use a mirror
near you, see @uref{https://www.gnu.org/order/ftp.html} for a full list.

@cindex Mailing lists
There are three public mailing lists of interest.  One for release
announcements, one for general questions and discussions about usage of the GMP
library and one for bug reports.  For more information, see

@display
@uref{https://gmplib.org/mailman/listinfo/}.
@end display

The proper place for bug reports is @email{gmp-bugs@@gmplib.org}.  See
@ref{Reporting Bugs} for information about reporting bugs.

@sp 1
@section How to use this Manual
@cindex About this manual

Everyone should read @ref{GMP Basics}.  If you need to install the library
yourself, then read @ref{Installing GMP}.  If you have a system with multiple
ABIs, then read @ref{ABI and ISA}, for the compiler options that must be used
on applications.

The rest of the manual can be used for later reference, although it is
probably a good idea to glance through it.


@node Installing GMP, GMP Basics, Introduction to GMP, Top
@comment  node-name,  next,  previous,  up
@chapter Installing GMP
@cindex Installing GMP
@cindex Configuring GMP
@cindex Building GMP

GMP has an autoconf/automake/libtool based configuration system.  On a
Unix-like system a basic build can be done with

@example
./configure
make
@end example

@noindent
Some self-tests can be run with

@example
make check
@end example

@noindent
And you can install (under @file{/usr/local} by default) with

@example
make install
@end example

If you experience problems, please report them to @email{gmp-bugs@@gmplib.org}.
See @ref{Reporting Bugs}, for information on what to include in useful bug
reports.

@menu
* Build Options::
* ABI and ISA::
* Notes for Package Builds::
* Notes for Particular Systems::
* Known Build Problems::
* Performance optimization::
@end menu


@node Build Options, ABI and ISA, Installing GMP, Installing GMP
@section Build Options
@cindex Build options

All the usual autoconf configure options are available, run @samp{./configure
--help} for a summary.  The file @file{INSTALL.autoconf} has some generic
installation information too.

@table @asis
@item Tools
@cindex Non-Unix systems
@samp{configure} requires various Unix-like tools.  See @ref{Notes for
Particular Systems}, for some options on non-Unix systems.

It might be possible to build without the help of @samp{configure}, certainly
all the code is there, but unfortunately you'll be on your own.

@item Build Directory
@cindex Build directory
To compile in a separate build directory, @command{cd} to that directory, and
prefix the configure command with the path to the GMP source directory.  For
example

@example
cd /my/build/dir
/my/sources/gmp-@value{VERSION}/configure
@end example

Not all @samp{make} programs have the necessary features (@code{VPATH}) to
support this.  In particular, SunOS and Slowaris @command{make} have bugs that
make them unable to build in a separate directory.  Use GNU @command{make}
instead.

@item @option{--prefix} and @option{--exec-prefix}
@cindex Prefix
@cindex Exec prefix
@cindex Install prefix
@cindex @code{--prefix}
@cindex @code{--exec-prefix}
The @option{--prefix} option can be used in the normal way to direct GMP to
install under a particular tree.  The default is @samp{/usr/local}.

@option{--exec-prefix} can be used to direct architecture-dependent files like
@file{libgmp.a} to a different location.  This can be used to share
architecture-independent parts like the documentation, but separate the
dependent parts.  Note however that @file{gmp.h} is
architecture-dependent since it encodes certain aspects of @file{libgmp}, so
it will be necessary to ensure both @file{$prefix/include} and
@file{$exec_prefix/include} are available to the compiler.

@item @option{--disable-shared}, @option{--disable-static}
@cindex @code{--disable-shared}
@cindex @code{--disable-static}
By default both shared and static libraries are built (where possible), but
one or other can be disabled.  Shared libraries result in smaller executables
and permit code sharing between separate running processes, but on some CPUs
are slightly slower, having a small cost on each function call.

@item Native Compilation, @option{--build=CPU-VENDOR-OS}
@cindex Native compilation
@cindex Build system
@cindex @code{--build}
For normal native compilation, the system can be specified with
@samp{--build}.  By default @samp{./configure} uses the output from running
@samp{./config.guess}.  On some systems @samp{./config.guess} can determine
the exact CPU type, on others it will be necessary to give it explicitly.  For
example,

@example
./configure --build=ultrasparc-sun-solaris2.7
@end example

In all cases the @samp{OS} part is important, since it controls how libtool
generates shared libraries.  Running @samp{./config.guess} is the simplest way
to see what it should be, if you don't know already.

@item Cross Compilation, @option{--host=CPU-VENDOR-OS}
@cindex Cross compiling
@cindex Host system
@cindex @code{--host}
When cross-compiling, the system used for compiling is given by @samp{--build}
and the system where the library will run is given by @samp{--host}.  For
example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries,

@example
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
@end example

Compiler tools are sought first with the host system type as a prefix.  For
example @command{m68k-mac-linux-gnu-ranlib} is tried, then plain
@command{ranlib}.  This makes it possible for a set of cross-compiling tools
to co-exist with native tools.  The prefix is the argument to @samp{--host},
and this can be an alias, such as @samp{m68k-linux}.  But note that tools
don't have to be setup this way, it's enough to just have a @env{PATH} with a
suitable cross-compiling @command{cc} etc.

Compiling for a different CPU in the same family as the build system is a form
of cross-compilation, though very possibly this would merely be special
options on a native compiler.  In any case @samp{./configure} avoids depending
on being able to run code on the build system, which is important when
creating binaries for a newer CPU since they very possibly won't run on the
build system.

In all cases the compiler must be able to produce an executable (of whatever
format) from a standard C @code{main}.  Although only object files will go to
make up @file{libgmp}, @samp{./configure} uses linking tests for various
purposes, such as determining what functions are available on the host system.

Currently a warning is given unless an explicit @samp{--build} is used when
cross-compiling, because it may not be possible to correctly guess the build
system type if the @env{PATH} has only a cross-compiling @command{cc}.

Note that the @samp{--target} option is not appropriate for GMP@.  It's for use
when building compiler tools, with @samp{--host} being where they will run,
and @samp{--target} what they'll produce code for.  Ordinary programs or
libraries like GMP are only interested in the @samp{--host} part, being where
they'll run.  (Some past versions of GMP used @samp{--target} incorrectly.)

@item CPU types
@cindex CPU types
In general, if you want a library that runs as fast as possible, you should
configure GMP for the exact CPU type your system uses.  However, this may mean
the binaries won't run on older members of the family, and might run slower on
other members, older or newer.  The best idea is always to build GMP for the
exact machine type you intend to run it on.

The following CPUs have specific support.  See @file{configure.ac} for details
of what code and compiler options they select.

@itemize @bullet

@c Keep this formatting, it's easy to read and it can be grepped to
@c automatically test that CPUs listed get through ./config.sub

@item
Alpha:
@nisamp{alpha},
@nisamp{alphaev5},
@nisamp{alphaev56},
@nisamp{alphapca56},
@nisamp{alphapca57},
@nisamp{alphaev6},
@nisamp{alphaev67},
@nisamp{alphaev68}
@nisamp{alphaev7}

@item
Cray:
@nisamp{c90},
@nisamp{j90},
@nisamp{t90},
@nisamp{sv1}

@item
HPPA:
@nisamp{hppa1.0},
@nisamp{hppa1.1},
@nisamp{hppa2.0},
@nisamp{hppa2.0n},
@nisamp{hppa2.0w},
@nisamp{hppa64}

@item
IA-64:
@nisamp{ia64},
@nisamp{itanium},
@nisamp{itanium2}

@item
MIPS:
@nisamp{mips},
@nisamp{mips3},
@nisamp{mips64}

@item
Motorola:
@nisamp{m68k},
@nisamp{m68000},
@nisamp{m68010},
@nisamp{m68020},
@nisamp{m68030},
@nisamp{m68040},
@nisamp{m68060},
@nisamp{m68302},
@nisamp{m68360},
@nisamp{m88k},
@nisamp{m88110}

@item
POWER:
@nisamp{power},
@nisamp{power1},
@nisamp{power2},
@nisamp{power2sc}

@item
PowerPC:
@nisamp{powerpc},
@nisamp{powerpc64},
@nisamp{powerpc401},
@nisamp{powerpc403},
@nisamp{powerpc405},
@nisamp{powerpc505},
@nisamp{powerpc601},
@nisamp{powerpc602},
@nisamp{powerpc603},
@nisamp{powerpc603e},
@nisamp{powerpc604},
@nisamp{powerpc604e},
@nisamp{powerpc620},
@nisamp{powerpc630},
@nisamp{powerpc740},
@nisamp{powerpc7400},
@nisamp{powerpc7450},
@nisamp{powerpc750},
@nisamp{powerpc801},
@nisamp{powerpc821},
@nisamp{powerpc823},
@nisamp{powerpc860},
@nisamp{powerpc970}

@item
SPARC:
@nisamp{sparc},
@nisamp{sparcv8},
@nisamp{microsparc},
@nisamp{supersparc},
@nisamp{sparcv9},
@nisamp{ultrasparc},
@nisamp{ultrasparc2},
@nisamp{ultrasparc2i},
@nisamp{ultrasparc3},
@nisamp{sparc64}

@item
x86 family:
@nisamp{i386},
@nisamp{i486},
@nisamp{i586},
@nisamp{pentium},
@nisamp{pentiummmx},
@nisamp{pentiumpro},
@nisamp{pentium2},
@nisamp{pentium3},
@nisamp{pentium4},
@nisamp{k6},
@nisamp{k62},
@nisamp{k63},
@nisamp{athlon},
@nisamp{amd64},
@nisamp{viac3},
@nisamp{viac32}

@item
Other:
@nisamp{arm},
@nisamp{sh},
@nisamp{sh2},
@nisamp{vax},
@end itemize

CPUs not listed will use generic C code.

@item Generic C Build
@cindex Generic C
If some of the assembly code causes problems, or if otherwise desired, the
generic C code can be selected with the configure @option{--disable-assembly}.

Note that this will run quite slowly, but it should be portable and should at
least make it possible to get something running if all else fails.

@item Fat binary, @option{--enable-fat}
@cindex Fat binary
@cindex @code{--enable-fat}
Using @option{--enable-fat} selects a ``fat binary'' build on x86, where
optimized low level subroutines are chosen at runtime according to the CPU
detected.  This means more code, but gives good performance on all x86 chips.
(This option might become available for more architectures in the future.)

@item @option{ABI}
@cindex ABI
On some systems GMP supports multiple ABIs (application binary interfaces),
meaning data type sizes and calling conventions.  By default GMP chooses the
best ABI available, but a particular ABI can be selected.  For example

@example
./configure --host=mips64-sgi-irix6 ABI=n32
@end example

See @ref{ABI and ISA}, for the available choices on relevant CPUs, and what
applications need to do.

@item @option{CC}, @option{CFLAGS}
@cindex C compiler
@cindex @code{CC}
@cindex @code{CFLAGS}
By default the C compiler used is chosen from among some likely candidates,
with @command{gcc} normally preferred if it's present.  The usual
@samp{CC=whatever} can be passed to @samp{./configure} to choose something
different.

For various systems, default compiler flags are set based on the CPU and
compiler.  The usual @samp{CFLAGS="-whatever"} can be passed to
@samp{./configure} to use something different or to set good flags for systems
GMP doesn't otherwise know.

The @samp{CC} and @samp{CFLAGS} used are printed during @samp{./configure},
and can be found in each generated @file{Makefile}.  This is the easiest way
to check the defaults when considering changing or adding something.

Note that when @samp{CC} and @samp{CFLAGS} are specified on a system
supporting multiple ABIs it's important to give an explicit
@samp{ABI=whatever}, since GMP can't determine the ABI just from the flags and
won't be able to select the correct assembly code.

If just @samp{CC} is selected then normal default @samp{CFLAGS} for that
compiler will be used (if GMP recognises it).  For example @samp{CC=gcc} can
be used to force the use of GCC, with default flags (and default ABI).

@item @option{CPPFLAGS}
@cindex @code{CPPFLAGS}
Any flags like @samp{-D} defines or @samp{-I} includes required by the
preprocessor should be set in @samp{CPPFLAGS} rather than @samp{CFLAGS}.
Compiling is done with both @samp{CPPFLAGS} and @samp{CFLAGS}, but
preprocessing uses just @samp{CPPFLAGS}.  This distinction is because most
preprocessors won't accept all the flags the compiler does.  Preprocessing is
done separately in some configure tests.

@item @option{CC_FOR_BUILD}
@cindex @code{CC_FOR_BUILD}
Some build-time programs are compiled and run to generate host-specific data
tables.  @samp{CC_FOR_BUILD} is the compiler used for this.  It doesn't need
to be in any particular ABI or mode, it merely needs to generate executables
that can run.  The default is to try the selected @samp{CC} and some likely
candidates such as @samp{cc} and @samp{gcc}, looking for something that works.

No flags are used with @samp{CC_FOR_BUILD} because a simple invocation like
@samp{cc foo.c} should be enough.  If some particular options are required
they can be included as for instance @samp{CC_FOR_BUILD="cc -whatever"}.

@item C++ Support, @option{--enable-cxx}
@cindex C++ support
@cindex @code{--enable-cxx}
C++ support in GMP can be enabled with @samp{--enable-cxx}, in which case a
C++ compiler will be required.  As a convenience @samp{--enable-cxx=detect}
can be used to enable C++ support only if a compiler can be found.  The C++
support consists of a library @file{libgmpxx.la} and header file
@file{gmpxx.h} (@pxref{Headers and Libraries}).

A separate @file{libgmpxx.la} has been adopted rather than having C++ objects
within @file{libgmp.la} in order to ensure dynamic linked C programs aren't
bloated by a dependency on the C++ standard library, and to avoid any chance
that the C++ compiler could be required when linking plain C programs.

@file{libgmpxx.la} will use certain internals from @file{libgmp.la} and can
only be expected to work with @file{libgmp.la} from the same GMP version.
Future changes to the relevant internals will be accompanied by renaming, so a
mismatch will cause unresolved symbols rather than perhaps mysterious
misbehaviour.

In general @file{libgmpxx.la} will be usable only with the C++ compiler that
built it, since name mangling and runtime support are usually incompatible
between different compilers.

@item @option{CXX}, @option{CXXFLAGS}
@cindex C++ compiler
@cindex @code{CXX}
@cindex @code{CXXFLAGS}
When C++ support is enabled, the C++ compiler and its flags can be set with
variables @samp{CXX} and @samp{CXXFLAGS} in the usual way.  The default for
@samp{CXX} is the first compiler that works from a list of likely candidates,
with @command{g++} normally preferred when available.  The default for
@samp{CXXFLAGS} is to try @samp{CFLAGS}, @samp{CFLAGS} without @samp{-g}, then
for @command{g++} either @samp{-g -O2} or @samp{-O2}, or for other compilers
@samp{-g} or nothing.  Trying @samp{CFLAGS} this way is convenient when using
@samp{gcc} and @samp{g++} together, since the flags for @samp{gcc} will
usually suit @samp{g++}.

It's important that the C and C++ compilers match, meaning their startup and
runtime support routines are compatible and that they generate code in the
same ABI (if there's a choice of ABIs on the system).  @samp{./configure}
isn't currently able to check these things very well itself, so for that
reason @samp{--disable-cxx} is the default, to avoid a build failure due to a
compiler mismatch.  Perhaps this will change in the future.

Incidentally, it's normally not good enough to set @samp{CXX} to the same as
@samp{CC}.  Although @command{gcc} for instance recognises @file{foo.cc} as
C++ code, only @command{g++} will invoke the linker the right way when
building an executable or shared library from C++ object files.

@item Temporary Memory, @option{--enable-alloca=<choice>}
@cindex Temporary memory
@cindex Stack overflow
@cindex @code{alloca}
@cindex @code{--enable-alloca}
GMP allocates temporary workspace using one of the following three methods,
which can be selected with for instance
@samp{--enable-alloca=malloc-reentrant}.

@itemize @bullet
@item
@samp{alloca} - C library or compiler builtin.
@item
@samp{malloc-reentrant} - the heap, in a re-entrant fashion.
@item
@samp{malloc-notreentrant} - the heap, with global variables.
@end itemize

For convenience, the following choices are also available.
@samp{--disable-alloca} is the same as @samp{no}.

@itemize @bullet
@item
@samp{yes} - a synonym for @samp{alloca}.
@item
@samp{no} - a synonym for @samp{malloc-reentrant}.
@item
@samp{reentrant} - @code{alloca} if available, otherwise
@samp{malloc-reentrant}.  This is the default.
@item
@samp{notreentrant} - @code{alloca} if available, otherwise
@samp{malloc-notreentrant}.
@end itemize

@code{alloca} is reentrant and fast, and is recommended.  It actually allocates
just small blocks on the stack; larger ones use malloc-reentrant.

@samp{malloc-reentrant} is, as the name suggests, reentrant and thread safe,
but @samp{malloc-notreentrant} is faster and should be used if reentrancy is
not required.

The two malloc methods in fact use the memory allocation functions selected by
@code{mp_set_memory_functions}, these being @code{malloc} and friends by
default.  @xref{Custom Allocation}.

An additional choice @samp{--enable-alloca=debug} is available, to help when
debugging memory related problems (@pxref{Debugging}).

@item FFT Multiplication, @option{--disable-fft}
@cindex FFT multiplication
@cindex @code{--disable-fft}
By default multiplications are done using Karatsuba, 3-way Toom, higher degree
Toom, and Fermat FFT@.  The FFT is only used on large to very large operands
and can be disabled to save code size if desired.

@item Assertion Checking, @option{--enable-assert}
@cindex Assertion checking
@cindex @code{--enable-assert}
This option enables some consistency checking within the library.  This can be
of use while debugging, @pxref{Debugging}.

@item Execution Profiling, @option{--enable-profiling=prof/gprof/instrument}
@cindex Execution profiling
@cindex @code{--enable-profiling}
Enable profiling support, in one of various styles, @pxref{Profiling}.

@item @option{MPN_PATH}
@cindex @code{MPN_PATH}
Various assembly versions of each mpn subroutines are provided.  For a given
CPU, a search is made though a path to choose a version of each.  For example
@samp{sparcv8} has

@example
MPN_PATH="sparc32/v8 sparc32 generic"
@end example

which means look first for v8 code, then plain sparc32 (which is v7), and
finally fall back on generic C@.  Knowledgeable users with special requirements
can specify a different path.  Normally this is completely unnecessary.

@item Documentation
@cindex Documentation formats
@cindex Texinfo
The source for the document you're now reading is @file{doc/gmp.texi}, in
Texinfo format, see @GMPreftop{texinfo, Texinfo}.

@cindex Postscript
@cindex DVI
@cindex PDF
Info format @samp{doc/gmp.info} is included in the distribution.  The usual
automake targets are available to make PostScript, DVI, PDF and HTML (these
will require various @TeX{} and Texinfo tools).

@cindex DocBook
@cindex XML
DocBook and XML can be generated by the Texinfo @command{makeinfo} program
too, see @ref{makeinfo options,, Options for @command{makeinfo}, texinfo,
Texinfo}.

Some supplementary notes can also be found in the @file{doc} subdirectory.

@end table


@need 2000
@node ABI and ISA, Notes for Package Builds, Build Options, Installing GMP
@section ABI and ISA
@cindex ABI
@cindex Application Binary Interface
@cindex ISA
@cindex Instruction Set Architecture

ABI (Application Binary Interface) refers to the calling conventions between
functions, meaning what registers are used and what sizes the various C data
types are.  ISA (Instruction Set Architecture) refers to the instructions and
registers a CPU has available.

Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the
latter for compatibility with older CPUs in the family.  GMP supports some
CPUs like this in both ABIs.  In fact within GMP @samp{ABI} means a
combination of chip ABI, plus how GMP chooses to use it.  For example in some
32-bit ABIs, GMP may support a limb as either a 32-bit @code{long} or a 64-bit
@code{long long}.

By default GMP chooses the best ABI available for a given system, and this
generally gives significantly greater speed.  But an ABI can be chosen
explicitly to make GMP compatible with other libraries, or particular
application requirements.  For example,

@example
./configure ABI=32
@end example

In all cases it's vital that all object code used in a given program is
compiled for the same ABI.

Usually a limb is implemented as a @code{long}.  When a @code{long long} limb
is used this is encoded in the generated @file{gmp.h}.  This is convenient for
applications, but it does mean that @file{gmp.h} will vary, and can't be just
copied around.  @file{gmp.h} remains compiler independent though, since all
compilers for a particular ABI will be expected to use the same limb type.

Currently no attempt is made to follow whatever conventions a system has for
installing library or header files built for a particular ABI@.  This will
probably only matter when installing multiple builds of GMP, and it might be
as simple as configuring with a special @samp{libdir}, or it might require
more than that.  Note that builds for different ABIs need to done separately,
with a fresh @command{./configure} and @command{make} each.

@sp 1
@table @asis
@need 1000
@item AMD64 (@samp{x86_64})
@cindex AMD64
On AMD64 systems supporting both 32-bit and 64-bit modes for applications, the
following ABI choices are available.

@table @asis
@item @samp{ABI=64}
The 64-bit ABI uses 64-bit limbs and pointers and makes full use of the chip
architecture.  This is the default.  Applications will usually not need
special compiler flags, but for reference the option is

@example
gcc  -m64
@end example

@item @samp{ABI=32}
The 32-bit ABI is the usual i386 conventions.  This will be slower, and is not
recommended except for inter-operating with other code not yet 64-bit capable.
Applications must be compiled with

@example
gcc  -m32
@end example

(In GCC 2.95 and earlier there's no @samp{-m32} option, it's the only mode.)

@item @samp{ABI=x32}
The x32 ABI uses 64-bit limbs but 32-bit pointers.  Like the 64-bit ABI, it
makes full use of the chip's arithmetic capabilities.  This ABI is not
supported by all operating systems.

@example
gcc  -mx32
@end example

@end table

@sp 1
@need 1000
@item HPPA 2.0 (@samp{hppa2.0*}, @samp{hppa64})
@cindex HPPA
@cindex HP-UX
@table @asis
@item @samp{ABI=2.0w}
The 2.0w ABI uses 64-bit limbs and pointers and is available on HP-UX 11 or
up.  Applications must be compiled with

@example
gcc [built for 2.0w]
cc  +DD64
@end example

@item @samp{ABI=2.0n}
The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal calling
conventions, but with 64-bit instructions permitted within functions.  GMP
uses a 64-bit @code{long long} for a limb.  This ABI is available on hppa64
GNU/Linux and on HP-UX 10 or higher.  Applications must be compiled with

@example
gcc [built for 2.0n]
cc  +DA2.0 +e
@end example

Note that current versions of GCC (eg.@: 3.2) don't generate 64-bit
instructions for @code{long long} operations and so may be slower than for
2.0w.  (The GMP assembly code is the same though.)

@item @samp{ABI=1.0}
HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit HPPA 1.0 ABI@.
No special compiler options are needed for applications.
@end table

All three ABIs are available for CPU types @samp{hppa2.0w}, @samp{hppa2.0} and
@samp{hppa64}, but for CPU type @samp{hppa2.0n} only 2.0n or 1.0 are
considered.

Note that GCC on HP-UX has no options to choose between 2.0n and 2.0w modes,
unlike HP @command{cc}.  Instead it must be built for one or the other ABI@.
GMP will detect how it was built, and skip to the corresponding @samp{ABI}.

@sp 1
@need 1500
@item IA-64 under HP-UX (@samp{ia64*-*-hpux*}, @samp{itanium*-*-hpux*})
@cindex IA-64
@cindex HP-UX
HP-UX supports two ABIs for IA-64.  GMP performance is the same in both.

@table @asis
@item @samp{ABI=32}
In the 32-bit ABI, pointers, @code{int}s and @code{long}s are 32 bits and GMP
uses a 64 bit @code{long long} for a limb.  Applications can be compiled
without any special flags since this ABI is the default in both HP C and GCC,
but for reference the flags are

@example
gcc  -milp32
cc   +DD32
@end example

@item @samp{ABI=64}
In the 64-bit ABI, @code{long}s and pointers are 64 bits and GMP uses a
@code{long} for a limb.  Applications must be compiled with

@example
gcc  -mlp64
cc   +DD64
@end example
@end table

On other IA-64 systems, GNU/Linux for instance, @samp{ABI=64} is the only
choice.

@sp 1
@need 1000
@item MIPS under IRIX 6 (@samp{mips*-*-irix[6789]})
@cindex MIPS
@cindex IRIX
IRIX 6 always has a 64-bit MIPS 3 or better CPU, and supports ABIs o32, n32,
and 64.  n32 or 64 are recommended, and GMP performance will be the same in
each.  The default is n32.

@table @asis
@item @samp{ABI=o32}
The o32 ABI is 32-bit pointers and integers, and no 64-bit operations.  GMP
will be slower than in n32 or 64, this option only exists to support old
compilers, eg.@: GCC 2.7.2.  Applications can be compiled with no special
flags on an old compiler, or on a newer compiler with

@example
gcc  -mabi=32
cc   -32
@end example

@item @samp{ABI=n32}
The n32 ABI is 32-bit pointers and integers, but with a 64-bit limb using a
@code{long long}.  Applications must be compiled with

@example
gcc  -mabi=n32
cc   -n32
@end example

@item @samp{ABI=64}
The 64-bit ABI is 64-bit pointers and integers.  Applications must be compiled
with

@example
gcc  -mabi=64
cc   -64
@end example
@end table

Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have the necessary
support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code.

@sp 1
@need 1000
@item PowerPC 64 (@samp{powerpc64}, @samp{powerpc620}, @samp{powerpc630}, @samp{powerpc970}, @samp{power4}, @samp{power5})
@cindex PowerPC
@table @asis
@item @samp{ABI=mode64}
@cindex AIX
The AIX 64 ABI uses 64-bit limbs and pointers and is the default on PowerPC 64
@samp{*-*-aix*} systems.  Applications must be compiled with

@example
gcc  -maix64
xlc  -q64
@end example

On 64-bit GNU/Linux, BSD, and Mac OS X/Darwin systems, the applications must
be compiled with

@example
gcc  -m64
@end example

@item @samp{ABI=mode32}
The @samp{mode32} ABI uses a 64-bit @code{long long} limb but with the chip
still in 32-bit mode and using 32-bit calling conventions.  This is the default
for systems where the true 64-bit ABI is unavailable.  No special compiler
options are typically needed for applications.  This ABI is not available under
AIX.

@item @samp{ABI=32}
This is the basic 32-bit PowerPC ABI, with a 32-bit limb.  No special compiler
options are needed for applications.
@end table

GMP's speed is greatest for the @samp{mode64} ABI, the @samp{mode32} ABI is 2nd
best.  In @samp{ABI=32} only the 32-bit ISA is used and this doesn't make full
use of a 64-bit chip.

@sp 1
@need 1000
@item Sparc V9 (@samp{sparc64}, @samp{sparcv9}, @samp{ultrasparc*})
@cindex Sparc V9
@cindex Solaris
@cindex Sun
@table @asis
@item @samp{ABI=64}
The 64-bit V9 ABI is available on the various BSD sparc64 ports, recent
versions of Sparc64 GNU/Linux, and Solaris 2.7 and up (when the kernel is in
64-bit mode).  GCC 3.2 or higher, or Sun @command{cc} is required.  On
GNU/Linux, depending on the default @command{gcc} mode, applications must be
compiled with

@example
gcc  -m64
@end example

On Solaris applications must be compiled with

@example
gcc  -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
cc   -xarch=v9
@end example

On the BSD sparc64 systems no special options are required, since 64-bits is
the only ABI available.

@item @samp{ABI=32}
For the basic 32-bit ABI, GMP still uses as much of the V9 ISA as it can.  In
the Sun documentation this combination is known as ``v8plus''.  On GNU/Linux,
depending on the default @command{gcc} mode, applications may need to be
compiled with

@example
gcc  -m32
@end example

On Solaris, no special compiler options are required for applications, though
using something like the following is recommended.  (@command{gcc} 2.8 and
earlier only support @samp{-mv8} though.)

@example
gcc  -mv8plus
cc   -xarch=v8plus
@end example
@end table

GMP speed is greatest in @samp{ABI=64}, so it's the default where available.
The speed is partly because there are extra registers available and partly
because 64-bits is considered the more important case and has therefore had
better code written for it.

Don't be confused by the names of the @samp{-m} and @samp{-x} compiler
options, they're called @samp{arch} but effectively control both ABI and ISA@.

On Solaris 2.6 and earlier, only @samp{ABI=32} is available since the kernel
doesn't save all registers.

On Solaris 2.7 with the kernel in 32-bit mode, a normal native build will
reject @samp{ABI=64} because the resulting executables won't run.
@samp{ABI=64} can still be built if desired by making it look like a
cross-compile, for example

@example
./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
@end example
@end table


@need 2000
@node Notes for Package Builds, Notes for Particular Systems, ABI and ISA, Installing GMP
@section Notes for Package Builds
@cindex Build notes for binary packaging
@cindex Packaged builds

GMP should present no great difficulties for packaging in a binary
distribution.

@cindex Libtool versioning
@cindex Shared library versioning
Libtool is used to build the library and @samp{-version-info} is set
appropriately, having started from @samp{3:0:0} in GMP 3.0 (@pxref{Versioning,
Library interface versions, Library interface versions, libtool, GNU
Libtool}).

The GMP 4 series will be upwardly binary compatible in each release and will
be upwardly binary compatible with all of the GMP 3 series.  Additional
function interfaces may be added in each release, so on systems where libtool
versioning is not fully checked by the loader an auxiliary mechanism may be
needed to express that a dynamic linked application depends on a new enough
GMP.

An auxiliary mechanism may also be needed to express that @file{libgmpxx.la}
(from @option{--enable-cxx}, @pxref{Build Options}) requires @file{libgmp.la}
from the same GMP version, since this is not done by the libtool versioning,
nor otherwise.  A mismatch will result in unresolved symbols from the linker,
or perhaps the loader.

When building a package for a CPU family, care should be taken to use
@samp{--host} (or @samp{--build}) to choose the least common denominator among
the CPUs which might use the package.  For example this might mean plain
@samp{sparc} (meaning V7) for SPARCs.

For x86s, @option{--enable-fat} sets things up for a fat binary build, making a
runtime selection of optimized low level routines.  This is a good choice for
packaging to run on a range of x86 chips.

Users who care about speed will want GMP built for their exact CPU type, to
make best use of the available optimizations.  Providing a way to suitably
rebuild a package may be useful.  This could be as simple as making it
possible for a user to omit @samp{--build} (and @samp{--host}) so
@samp{./config.guess} will detect the CPU@.  But a way to manually specify a
@samp{--build} will be wanted for systems where @samp{./config.guess} is
inexact.

On systems with multiple ABIs, a packaged build will need to decide which
among the choices is to be provided, see @ref{ABI and ISA}.  A given run of
@samp{./configure} etc will only build one ABI@.  If a second ABI is also
required then a second run of @samp{./configure} etc must be made, starting
from a clean directory tree (@samp{make distclean}).

As noted under ``ABI and ISA'', currently no attempt is made to follow system
conventions for install locations that vary with ABI, such as
@file{/usr/lib/sparcv9} for @samp{ABI=64} as opposed to @file{/usr/lib} for
@samp{ABI=32}.  A package build can override @samp{libdir} and other standard
variables as necessary.

Note that @file{gmp.h} is a generated file, and will be architecture and ABI
dependent.  When attempting to install two ABIs simultaneously it will be
important that an application compile gets the correct @file{gmp.h} for its
desired ABI@.  If compiler include paths don't vary with ABI options then it
might be necessary to create a @file{/usr/include/gmp.h} which tests
preprocessor symbols and chooses the correct actual @file{gmp.h}.


@need 2000
@node Notes for Particular Systems, Known Build Problems, Notes for Package Builds, Installing GMP
@section Notes for Particular Systems
@cindex Build notes for particular systems
@cindex Particular systems
@cindex Systems
@table @asis

@c This section is more or less meant for notes about performance or about
@c build problems that have been worked around but might leave a user
@c scratching their head.  Fun with different ABIs on a system belongs in the
@c above section.

@item AIX 3 and 4
@cindex AIX
On systems @samp{*-*-aix[34]*} shared libraries are disabled by default, since
some versions of the native @command{ar} fail on the convenience libraries
used.  A shared build can be attempted with

@example
./configure --enable-shared --disable-static
@end example

Note that the @samp{--disable-static} is necessary because in a shared build
libtool makes @file{libgmp.a} a symlink to @file{libgmp.so}, apparently for
the benefit of old versions of @command{ld} which only recognise @file{.a},
but unfortunately this is done even if a fully functional @command{ld} is
available.

@item ARM
@cindex ARM
On systems @samp{arm*-*-*}, versions of GCC up to and including 2.95.3 have a
bug in unsigned division, giving wrong results for some operands.  GMP
@samp{./configure} will demand GCC 2.95.4 or later.

@item Compaq C++
@cindex Compaq C++
Compaq C++ on OSF 5.1 has two flavours of @code{iostream}, a standard one and
an old pre-standard one (see @samp{man iostream_intro}).  GMP can only use the
standard one, which unfortunately is not the default but must be selected by
defining @code{__USE_STD_IOSTREAM}.  Configure with for instance

@example
./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
@end example

@item Floating Point Mode
@cindex Floating point mode
@cindex Hardware floating point mode
@cindex Precision of hardware floating point
@cindex x87
On some systems, the hardware floating point has a control mode which can set
all operations to be done in a particular precision, for instance single,
double or extended on x86 systems (x87 floating point).  The GMP functions
involving a @code{double} cannot be expected to operate to their full
precision when the hardware is in single precision mode.  Of course this
affects all code, including application code, not just GMP.

@item FreeBSD 7.x, 8.x, 9.0, 9.1, 9.2
@cindex FreeBSD
@command{m4} in these releases of FreeBSD has an eval function which ignores
its 2nd and 3rd arguments, which makes it unsuitable for @file{.asm} file
processing.  @samp{./configure} will detect the problem and either abort or
choose another m4 in the @env{PATH}.  The bug is fixed in FreeBSD 9.3 and 10.0,
so either upgrade or use GNU m4.  Note that the FreeBSD package system installs
GNU m4 under the name @samp{gm4}, which GMP cannot guess.

@item FreeBSD 7.x, 8.x, 9.x
@cindex FreeBSD
GMP releases starting with 6.0 do not support @samp{ABI=32} on FreeBSD/amd64
prior to release 10.0 of the system.  The cause is a broken @code{limits.h},
which GMP no longer works around.

@item MS-DOS and MS Windows
@cindex MS-DOS
@cindex MS Windows
@cindex Windows
@cindex Cygwin
@cindex DJGPP
@cindex MINGW
On an MS-DOS system DJGPP can be used to build GMP, and on an MS Windows
system Cygwin, DJGPP and MINGW can be used.  All three are excellent ports of
GCC and the various GNU tools.

@display
@uref{https://www.cygwin.com/}
@uref{http://www.delorie.com/djgpp/}
@uref{http://www.mingw.org/}
@end display

@cindex Interix
@cindex Services for Unix
Microsoft also publishes an Interix ``Services for Unix'' which can be used to
build GMP on Windows (with a normal @samp{./configure}), but it's not free
software.

@item MS Windows DLLs
@cindex DLLs
@cindex MS Windows
@cindex Windows
On systems @samp{*-*-cygwin*}, @samp{*-*-mingw*} and @samp{*-*-pw32*} by
default GMP builds only a static library, but a DLL can be built instead using

@example
./configure --disable-static --enable-shared
@end example

Static and DLL libraries can't both be built, since certain export directives
in @file{gmp.h} must be different.

A MINGW DLL build of GMP can be used with Microsoft C@.  Libtool doesn't
install a @file{.lib} format import library, but it can be created with MS
@command{lib} as follows, and copied to the install directory.  Similarly for
@file{libmp} and @file{libgmpxx}.

@example
cd .libs
lib /def:libgmp-3.dll.def /out:libgmp-3.lib
@end example

MINGW uses the C runtime library @samp{msvcrt.dll} for I/O, so applications
wanting to use the GMP I/O routines must be compiled with @samp{cl /MD} to do
the same.  If one of the other C runtime library choices provided by MS C is
desired then the suggestion is to use the GMP string functions and confine I/O
to the application.

@item Motorola 68k CPU Types
@cindex 68000
@samp{m68k} is taken to mean 68000.  @samp{m68020} or higher will give a
performance boost on applicable CPUs.  @samp{m68360} can be used for CPU32
series chips.  @samp{m68302} can be used for ``Dragonball'' series chips,
though this is merely a synonym for @samp{m68000}.

@item NetBSD 5.x
@cindex NetBSD
@command{m4} in these releases of NetBSD has an eval function which ignores its
2nd and 3rd arguments, which makes it unsuitable for @file{.asm} file
processing.  @samp{./configure} will detect the problem and either abort or
choose another m4 in the @env{PATH}.  The bug is fixed in NetBSD 6, so either
upgrade or use GNU m4.  Note that the NetBSD package system installs GNU m4
under the name @samp{gm4}, which GMP cannot guess.

@item OpenBSD 2.6
@cindex OpenBSD
@command{m4} in this release of OpenBSD has a bug in @code{eval} that makes it
unsuitable for @file{.asm} file processing.  @samp{./configure} will detect
the problem and either abort or choose another m4 in the @env{PATH}.  The bug
is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.

@item Power CPU Types
@cindex Power/PowerPC
In GMP, CPU types @samp{power*} and @samp{powerpc*} will each use instructions
not available on the other, so it's important to choose the right one for the
CPU that will be used.  Currently GMP has no assembly code support for using
just the common instruction subset.  To get executables that run on both, the
current suggestion is to use the generic C code (@option{--disable-assembly}),
possibly with appropriate compiler options (like @samp{-mcpu=common} for
@command{gcc}).  CPU @samp{rs6000} (which is not a CPU but a family of
workstations) is accepted by @file{config.sub}, but is currently equivalent to
@option{--disable-assembly}.

@item Sparc CPU Types
@cindex Sparc
@samp{sparcv8} or @samp{supersparc} on relevant systems will give a
significant performance increase over the V7 code selected by plain
@samp{sparc}.

@item Sparc App Regs
@cindex Sparc
The GMP assembly code for both 32-bit and 64-bit Sparc clobbers the
``application registers'' @code{g2}, @code{g3} and @code{g4}, the same way
that the GCC default @samp{-mapp-regs} does (@pxref{SPARC Options,, SPARC
Options, gcc, Using the GNU Compiler Collection (GCC)}).

This makes that code unsuitable for use with the special V9
@samp{-mcmodel=embmedany} (which uses @code{g4} as a data segment pointer), and
for applications wanting to use those registers for special purposes.  In these
cases the only suggestion currently is to build GMP with
@option{--disable-assembly} to avoid the assembly code.

@item SunOS 4
@cindex SunOS
@command{/usr/bin/m4} lacks various features needed to process @file{.asm}
files, and instead @samp{./configure} will automatically use
@command{/usr/5bin/m4}, which we believe is always available (if not then use
GNU m4).

@item x86 CPU Types
@cindex x86
@cindex 80x86
@cindex i386
@samp{i586}, @samp{pentium} or @samp{pentiummmx} code is good for its intended
P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II,
P-III)@.  @samp{i386} is a better choice when making binaries that must run on
both.

@item x86 MMX and SSE2 Code
@cindex MMX
@cindex SSE2
If the CPU selected has MMX code but the assembler doesn't support it, a
warning is given and non-MMX code is used instead.  This will be an inferior
build, since the MMX code that's present is there because it's faster than the
corresponding plain integer code.  The same applies to SSE2.

Old versions of @samp{gas} don't support MMX instructions, in particular
version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent OpenBSD 3.1
doesn't.

Solaris 2.6 and 2.7 @command{as} generate incorrect object code for register
to register @code{movq} instructions, and so can't be used for MMX code.
Install a recent @command{gas} if MMX code is wanted on these systems.
@end table


@need 2000
@node Known Build Problems, Performance optimization, Notes for Particular Systems, Installing GMP
@section Known Build Problems
@cindex Build problems known

@c This section is more or less meant for known build problems that are not
@c otherwise worked around and require some sort of manual intervention.

You might find more up-to-date information at @uref{https://gmplib.org/}.

@table @asis
@item Compiler link options
The version of libtool currently in use rather aggressively strips compiler
options when linking a shared library.  This will hopefully be relaxed in the
future, but for now if this is a problem the suggestion is to create a little
script to hide them, and for instance configure with

@example
./configure CC=gcc-with-my-options
@end example

@item DJGPP (@samp{*-*-msdosdjgpp*})
@cindex DJGPP
The DJGPP port of @command{bash} 2.03 is unable to run the @samp{configure}
script, it exits silently, having died writing a preamble to
@file{config.log}.  Use @command{bash} 2.04 or higher.

@samp{make all} was found to run out of memory during the final
@file{libgmp.la} link on one system tested, despite having 64Mb available.
Running @samp{make libgmp.la} directly helped, perhaps recursing into the
various subdirectories uses up memory.

@item GNU binutils @command{strip} prior to 2.12
@cindex Stripped libraries
@cindex Binutils @command{strip}
@cindex GNU @command{strip}
@command{strip} from GNU binutils 2.11 and earlier should not be used on the
static libraries @file{libgmp.a} and @file{libmp.a} since it will discard all
but the last of multiple archive members with the same name, like the three
versions of @file{init.o} in @file{libgmp.a}.  Binutils 2.12 or higher can be
used successfully.

The shared libraries @file{libgmp.so} and @file{libmp.so} are not affected by
this and any version of @command{strip} can be used on them.

@item @command{make} syntax error
@cindex SCO
@cindex IRIX
On certain versions of SCO OpenServer 5 and IRIX 6.5 the native @command{make}
is unable to handle the long dependencies list for @file{libgmp.la}.  The
symptom is a ``syntax error'' on the following line of the top-level
@file{Makefile}.

@example
libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)
@end example

Either use GNU Make, or as a workaround remove
@code{$(libgmp_la_DEPENDENCIES)} from that line (which will make the initial
build work, but if any recompiling is done @file{libgmp.la} might not be
rebuilt).

@item MacOS X (@samp{*-*-darwin*})
@cindex MacOS X
@cindex Darwin
Libtool currently only knows how to create shared libraries on MacOS X using
the native @command{cc} (which is a modified GCC), not a plain GCC@.  A
static-only build should work though (@samp{--disable-shared}).

@item NeXT prior to 3.3
@cindex NeXT
The system compiler on old versions of NeXT was a massacred and old GCC, even
if it called itself @file{cc}.  This compiler cannot be used to build GMP, you
need to get a real GCC, and install that.  (NeXT may have fixed this in
release 3.3 of their system.)

@item POWER and PowerPC
@cindex Power/PowerPC
Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP on POWER or
PowerPC@.  If you want to use GCC for these machines, get GCC 2.7.2.1 (or
later).

@item Sequent Symmetry
@cindex Sequent Symmetry
Use the GNU assembler instead of the system assembler, since the latter has
serious bugs.

@item Solaris 2.6
@cindex Solaris
The system @command{sed} prints an error ``Output line too long'' when libtool
builds @file{libgmp.la}.  This doesn't seem to cause any obvious ill effects,
but GNU @command{sed} is recommended, to avoid any doubt.

@item Sparc Solaris 2.7 with gcc 2.95.2 in @samp{ABI=32}
@cindex Solaris
A shared library build of GMP seems to fail in this combination, it builds but
then fails the tests, apparently due to some incorrect data relocations within
@code{gmp_randinit_lc_2exp_size}.  The exact cause is unknown,
@samp{--disable-shared} is recommended.
@end table


@need 2000
@node Performance optimization, , Known Build Problems, Installing GMP
@section Performance optimization
@cindex Optimizing performance

@c At some point, this should perhaps move to a separate chapter on optimizing
@c performance.

For optimal performance, build GMP for the exact CPU type of the target
computer, see @ref{Build Options}.

Unlike what is the case for most other programs, the compiler typically
doesn't matter much, since GMP uses assembly language for the most critical
operation.

In particular for long-running GMP applications, and applications demanding
extremely large numbers, building and running the @code{tuneup} program in the
@file{tune} subdirectory, can be important.  For example,

@example
cd tune
make tuneup
./tuneup
@end example

will generate better contents for the @file{gmp-mparam.h} parameter file.

To use the results, put the output in the file indicated in the
@samp{Parameters for ...} header.  Then recompile from scratch.

The @code{tuneup} program takes one useful parameter, @samp{-f NNN}, which
instructs the program how long to check FFT multiply parameters.  If you're
going to use GMP for extremely large numbers, you may want to run @code{tuneup}
with a large NNN value.


@node GMP Basics, Reporting Bugs, Installing GMP, Top
@comment  node-name,  next,  previous,  up
@chapter GMP Basics
@cindex Basics

@strong{Using functions, macros, data types, etc.@: not documented in this
manual is strongly discouraged.  If you do so your application is guaranteed
to be incompatible with future versions of GMP.}

@menu
* Headers and Libraries::
* Nomenclature and Types::
* Function Classes::
* Variable Conventions::
* Parameter Conventions::
* Memory Management::
* Reentrancy::
* Useful Macros and Constants::
* Compatibility with older versions::
* Demonstration Programs::
* Efficiency::
* Debugging::
* Profiling::
* Autoconf::
* Emacs::
@end menu

@node Headers and Libraries, Nomenclature and Types, GMP Basics, GMP Basics
@section Headers and Libraries
@cindex Headers

@cindex @file{gmp.h}
@cindex Include files
@cindex @code{#include}
All declarations needed to use GMP are collected in the include file
@file{gmp.h}.  It is designed to work with both C and C++ compilers.

@example
#include <gmp.h>
@end example

@cindex @code{stdio.h}
Note however that prototypes for GMP functions with @code{FILE *} parameters
are only provided if @code{<stdio.h>} is included too.

@example
#include <stdio.h>
#include <gmp.h>
@end example

@cindex @code{stdarg.h}
Likewise @code{<stdarg.h>} is required for prototypes with @code{va_list}
parameters, such as @code{gmp_vprintf}.  And @code{<obstack.h>} for prototypes
with @code{struct obstack} parameters, such as @code{gmp_obstack_printf}, when
available.

@cindex Libraries
@cindex Linking
@cindex @code{libgmp}
All programs using GMP must link against the @file{libgmp} library.  On a
typical Unix-like system this can be done with @samp{-lgmp}, for example

@example
gcc myprogram.c -lgmp
@end example

@cindex @code{libgmpxx}
GMP C++ functions are in a separate @file{libgmpxx} library.  This is built
and installed if C++ support has been enabled (@pxref{Build Options}).  For
example,

@example
g++ mycxxprog.cc -lgmpxx -lgmp
@end example

@cindex Libtool
GMP is built using Libtool and an application can use that to link if desired,
@GMPpxreftop{libtool, GNU Libtool}.

If GMP has been installed to a non-standard location then it may be necessary
to use @samp{-I} and @samp{-L} compiler options to point to the right
directories, and some sort of run-time path for a shared library.


@node Nomenclature and Types, Function Classes, Headers and Libraries, GMP Basics
@section Nomenclature and Types
@cindex Nomenclature
@cindex Types

@cindex Integer
@tindex @code{mpz_t}
In this manual, @dfn{integer} usually means a multiple precision integer, as
defined by the GMP library.  The C data type for such integers is @code{mpz_t}.
Here are some examples of how to declare such integers:

@example
mpz_t sum;

struct foo @{ mpz_t x, y; @};

mpz_t vec[20];
@end example

@cindex Rational number
@tindex @code{mpq_t}
@dfn{Rational number} means a multiple precision fraction.  The C data type
for these fractions is @code{mpq_t}.  For example:

@example
mpq_t quotient;
@end example

@cindex Floating-point number
@tindex @code{mpf_t}
@dfn{Floating point number} or @dfn{Float} for short, is an arbitrary precision
mantissa with a limited precision exponent.  The C data type for such objects
is @code{mpf_t}.  For example:

@example
mpf_t fp;
@end example

@tindex @code{mp_exp_t}
The floating point functions accept and return exponents in the C type
@code{mp_exp_t}.  Currently this is usually a @code{long}, but on some systems
it's an @code{int} for efficiency.

@cindex Limb
@tindex @code{mp_limb_t}
A @dfn{limb} means the part of a multi-precision number that fits in a single
machine word.  (We chose this word because a limb of the human body is
analogous to a digit, only larger, and containing several digits.)  Normally a
limb is 32 or 64 bits.  The C data type for a limb is @code{mp_limb_t}.

@tindex @code{mp_size_t}
Counts of limbs of a multi-precision number represented in the C type
@code{mp_size_t}.  Currently this is normally a @code{long}, but on some
systems it's an @code{int} for efficiency, and on some systems it will be
@code{long long} in the future.

@tindex @code{mp_bitcnt_t}
Counts of bits of a multi-precision number are represented in the C type
@code{mp_bitcnt_t}.  Currently this is always an @code{unsigned long}, but on
some systems it will be an @code{unsigned long long} in the future.

@cindex Random state
@tindex @code{gmp_randstate_t}
@dfn{Random state} means an algorithm selection and current state data.  The C
data type for such objects is @code{gmp_randstate_t}.  For example:

@example
gmp_randstate_t rstate;
@end example

Also, in general @code{mp_bitcnt_t} is used for bit counts and ranges, and
@code{size_t} is used for byte or character counts.


@node Function Classes, Variable Conventions, Nomenclature and Types, GMP Basics
@section Function Classes
@cindex Function classes

There are six classes of functions in the GMP library:

@enumerate
@item
Functions for signed integer arithmetic, with names beginning with
@code{mpz_}.  The associated type is @code{mpz_t}.  There are about 150
functions in this class.  (@pxref{Integer Functions})

@item
Functions for rational number arithmetic, with names beginning with
@code{mpq_}.  The associated type is @code{mpq_t}.  There are about 35
functions in this class, but the integer functions can be used for arithmetic
on the numerator and denominator separately.  (@pxref{Rational Number
Functions})

@item
Functions for floating-point arithmetic, with names beginning with
@code{mpf_}.  The associated type is @code{mpf_t}.  There are about 70
functions is this class.  (@pxref{Floating-point Functions})

@item
Fast low-level functions that operate on natural numbers.  These are used by
the functions in the preceding groups, and you can also call them directly
from very time-critical user programs.  These functions' names begin with
@code{mpn_}.  The associated type is array of @code{mp_limb_t}.  There are
about 60 (hard-to-use) functions in this class.  (@pxref{Low-level Functions})

@item
Miscellaneous functions.  Functions for setting up custom allocation and
functions for generating random numbers.  (@pxref{Custom Allocation}, and
@pxref{Random Number Functions})
@end enumerate


@node Variable Conventions, Parameter Conventions, Function Classes, GMP Basics
@section Variable Conventions
@cindex Variable conventions
@cindex Conventions for variables

GMP functions generally have output arguments before input arguments.  This
notation is by analogy with the assignment operator.

GMP lets you use the same variable for both input and output in one call.  For
example, the main function for integer multiplication, @code{mpz_mul}, can be
used to square @code{x} and put the result back in @code{x} with

@example
mpz_mul (x, x, x);
@end example

Before you can assign to a GMP variable, you need to initialize it by calling
one of the special initialization functions.  When you're done with a
variable, you need to clear it out, using one of the functions for that
purpose.  Which function to use depends on the type of variable.  See the
chapters on integer functions, rational number functions, and floating-point
functions for details.

A variable should only be initialized once, or at least cleared between each
initialization.  After a variable has been initialized, it may be assigned to
any number of times.

For efficiency reasons, avoid excessive initializing and clearing.  In
general, initialize near the start of a function and clear near the end.  For
example,

@example
void
foo (void)
@{
  mpz_t  n;
  int    i;
  mpz_init (n);
  for (i = 1; i < 100; i++)
    @{
      mpz_mul (n, @dots{});
      mpz_fdiv_q (n, @dots{});
      @dots{}
    @}
  mpz_clear (n);
@}
@end example

GMP types like @code{mpz_t} are implemented as one-element arrays of certain
structures.  Declaring a variable creates an object with the fields GMP needs,
but variables are normally manipulated by using the pointer to the object.  For
both behavior and efficiency reasons, it is discouraged to make copies of the
GMP object itself (either directly or via aggregate objects containing such GMP
objects).  If copies are done, all of them must be used read-only; using a copy
as the output of some function will invalidate all the other copies.  Note that
the actual fields in each @code{mpz_t} etc are for internal use only and should
not be accessed directly by code that expects to be compatible with future GMP
releases.

@node Parameter Conventions, Memory Management, Variable Conventions, GMP Basics
@section Parameter Conventions
@cindex Parameter conventions
@cindex Conventions for parameters

When a GMP variable is used as a function parameter, it's effectively a
call-by-reference, meaning that when the function stores a value there it will
change the original in the caller.  Parameters which are input-only can be
designated @code{const} to provoke a compiler error or warning on attempting to
modify them.

When a function is going to return a GMP result, it should designate a
parameter that it sets, like the library functions do.  More than one value
can be returned by having more than one output parameter, again like the
library functions.  A @code{return} of an @code{mpz_t} etc doesn't return the
object, only a pointer, and this is almost certainly not what's wanted.

Here's an example accepting an @code{mpz_t} parameter, doing a calculation,
and storing the result to the indicated parameter.

@example
void
foo (mpz_t result, const mpz_t param, unsigned long n)
@{
  unsigned long  i;
  mpz_mul_ui (result, param, n);
  for (i = 1; i < n; i++)
    mpz_add_ui (result, result, i*7);
@}

int
main (void)
@{
  mpz_t  r, n;
  mpz_init (r);
  mpz_init_set_str (n, "123456", 0);
  foo (r, n, 20L);
  gmp_printf ("%Zd\n", r);
  return 0;
@}
@end example

Our function @code{foo} works even if its caller passes the same variable for
@code{param} and @code{result}, just like the library functions.  But
sometimes it's tricky to make that work, and an application might not want to
bother supporting that sort of thing.

Since GMP types are implemented as one-element arrays, using a GMP variable as
a parameter passes a pointer to the object. Hence the call-by-reference.


@need 1000
@node Memory Management, Reentrancy, Parameter Conventions, GMP Basics
@section Memory Management
@cindex Memory management

The GMP types like @code{mpz_t} are small, containing only a couple of sizes,
and pointers to allocated data.  Once a variable is initialized, GMP takes
care of all space allocation.  Additional space is allocated whenever a
variable doesn't have enough.

@code{mpz_t} and @code{mpq_t} variables never reduce their allocated space.
Normally this is the best policy, since it avoids frequent reallocation.
Applications that need to return memory to the heap at some particular point
can use @code{mpz_realloc2}, or clear variables no longer needed.

@code{mpf_t} variables, in the current implementation, use a fixed amount of
space, determined by the chosen precision and allocated at initialization, so
their size doesn't change.

All memory is allocated using @code{malloc} and friends by default, but this
can be changed, see @ref{Custom Allocation}.  Temporary memory on the stack is
also used (via @code{alloca}), but this can be changed at build-time if
desired, see @ref{Build Options}.


@node Reentrancy, Useful Macros and Constants, Memory Management, GMP Basics
@section Reentrancy
@cindex Reentrancy
@cindex Thread safety
@cindex Multi-threading

@noindent
GMP is reentrant and thread-safe, with some exceptions:

@itemize @bullet
@item
If configured with @option{--enable-alloca=malloc-notreentrant} (or with
@option{--enable-alloca=notreentrant} when @code{alloca} is not available),
then naturally GMP is not reentrant.

@item
@code{mpf_set_default_prec} and @code{mpf_init} use a global variable for the
selected precision.  @code{mpf_init2} can be used instead, and in the C++
interface an explicit precision to the @code{mpf_class} constructor.

@item
@code{mpz_random} and the other old random number functions use a global
random state and are hence not reentrant.  The newer random number functions
that accept a @code{gmp_randstate_t} parameter can be used instead.

@item
@code{gmp_randinit} (obsolete) returns an error indication through a global
variable, which is not thread safe.  Applications are advised to use
@code{gmp_randinit_default} or @code{gmp_randinit_lc_2exp} instead.

@item
@code{mp_set_memory_functions} uses global variables to store the selected
memory allocation functions.

@item
If the memory allocation functions set by a call to
@code{mp_set_memory_functions} (or @code{malloc} and friends by default) are
not reentrant, then GMP will not be reentrant either.

@item
If the standard I/O functions such as @code{fwrite} are not reentrant then the
GMP I/O functions using them will not be reentrant either.

@item
It's safe for two threads to read from the same GMP variable simultaneously,
but it's not safe for one to read while another might be writing, nor for
two threads to write simultaneously.  It's not safe for two threads to
generate a random number from the same @code{gmp_randstate_t} simultaneously,
since this involves an update of that variable.
@end itemize


@need 2000
@node Useful Macros and Constants, Compatibility with older versions, Reentrancy, GMP Basics
@section Useful Macros and Constants
@cindex Useful macros and constants
@cindex Constants

@deftypevr {Global Constant} {const int} mp_bits_per_limb
@findex mp_bits_per_limb
@cindex Bits per limb
@cindex Limb size
The number of bits per limb.
@end deftypevr

@defmac __GNU_MP_VERSION
@defmacx __GNU_MP_VERSION_MINOR
@defmacx __GNU_MP_VERSION_PATCHLEVEL
@cindex Version number
@cindex GMP version number
The major and minor GMP version, and patch level, respectively, as integers.
For GMP i.j, these numbers will be i, j, and 0, respectively.
For GMP i.j.k, these numbers will be i, j, and k, respectively.
@end defmac

@deftypevr {Global Constant} {const char * const} gmp_version
@findex gmp_version
The GMP version number, as a null-terminated string, in the form ``i.j.k''.
This release is @nicode{"@value{VERSION}"}.  Note that the format ``i.j'' was
used, before version 4.3.0, when k was zero.
@end deftypevr

@defmac __GMP_CC
@defmacx __GMP_CFLAGS
The compiler and compiler flags, respectively, used when compiling GMP, as
strings.
@end defmac


@node Compatibility with older versions, Demonstration Programs, Useful Macros and Constants, GMP Basics
@section Compatibility with older versions
@cindex Compatibility with older versions
@cindex Past GMP versions
@cindex Upward compatibility

This version of GMP is upwardly binary compatible with all 5.x, 4.x, and 3.x
versions, and upwardly compatible at the source level with all 2.x versions,
with the following exceptions.

@itemize @bullet
@item
@code{mpn_gcd} had its source arguments swapped as of GMP 3.0, for consistency
with other @code{mpn} functions.

@item
@code{mpf_get_prec} counted precision slightly differently in GMP 3.0 and
3.0.1, but in 3.1 reverted to the 2.x style.

@item
@code{mpn_bdivmod}, documented as preliminary in GMP 4, has been removed.
@end itemize

There are a number of compatibility issues between GMP 1 and GMP 2 that of
course also apply when porting applications from GMP 1 to GMP 5.  Please
see the GMP 2 manual for details.

@c @item Integer division functions round the result differently.  The obsolete
@c functions (@code{mpz_div}, @code{mpz_divmod}, @code{mpz_mdiv},
@c @code{mpz_mdivmod}, etc) now all use floor rounding (i.e., they round the
@c quotient towards
@c @ifinfo
@c @minus{}infinity).
@c @end ifinfo
@c @iftex
@c @tex
@c $-\infty$).
@c @end tex
@c @end iftex
@c There are a lot of functions for integer division, giving the user better
@c control over the rounding.

@c @item The function @code{mpz_mod} now compute the true @strong{mod} function.

@c @item The functions @code{mpz_powm} and @code{mpz_powm_ui} now use
@c @strong{mod} for reduction.

@c @item The assignment functions for rational numbers do no longer canonicalize
@c their results.  In the case a non-canonical result could arise from an
@c assignment, the user need to insert an explicit call to
@c @code{mpq_canonicalize}.  This change was made for efficiency.

@c @item Output generated by @code{mpz_out_raw} in this release cannot be read
@c by @code{mpz_inp_raw} in previous releases.  This change was made for making
@c the file format truly portable between machines with different word sizes.

@c @item Several @code{mpn} functions have changed.  But they were intentionally
@c undocumented in previous releases.

@c @item The functions @code{mpz_cmp_ui}, @code{mpz_cmp_si}, and @code{mpq_cmp_ui}
@c are now implemented as macros, and thereby sometimes evaluate their
@c arguments multiple times.

@c @item The functions @code{mpz_pow_ui} and @code{mpz_ui_pow_ui} now yield 1
@c for 0^0.  (In version 1, they yielded 0.)

@c In version 1 of the library, @code{mpq_set_den} handled negative
@c denominators by copying the sign to the numerator.  That is no longer done.

@c Pure assignment functions do not canonicalize the assigned variable.  It is
@c the responsibility of the user to canonicalize the assigned variable before
@c any arithmetic operations are performed on that variable.
@c Note that this is an incompatible change from version 1 of the library.

@c @end enumerate


@need 1000
@node Demonstration Programs, Efficiency, Compatibility with older versions, GMP Basics
@section Demonstration programs
@cindex Demonstration programs
@cindex Example programs
@cindex Sample programs
The @file{demos} subdirectory has some sample programs using GMP@.  These
aren't built or installed, but there's a @file{Makefile} with rules for them.
For instance,

@example
make pexpr
./pexpr 68^975+10
@end example

@noindent
The following programs are provided

@itemize @bullet
@item
@cindex Expression parsing demo
@cindex Parsing expressions demo
@samp{pexpr} is an expression evaluator, the program used on the GMP web page.
@item
@cindex Expression parsing demo
@cindex Parsing expressions demo
The @samp{calc} subdirectory has a similar but simpler evaluator using
@command{lex} and @command{yacc}.
@item
@cindex Expression parsing demo
@cindex Parsing expressions demo
The @samp{expr} subdirectory is yet another expression evaluator, a library
designed for ease of use within a C program.  See @file{demos/expr/README} for
more information.
@item
@cindex Factorization demo
@samp{factorize} is a Pollard-Rho factorization program.
@item
@samp{isprime} is a command-line interface to the @code{mpz_probab_prime_p}
function.
@item
@samp{primes} counts or lists primes in an interval, using a sieve.
@item
@samp{qcn} is an example use of @code{mpz_kronecker_ui} to estimate quadratic
class numbers.
@item
@cindex @code{perl}
@cindex GMP Perl module
@cindex Perl module
The @samp{perl} subdirectory is a comprehensive perl interface to GMP@.  See
@file{demos/perl/INSTALL} for more information.  Documentation is in POD
format in @file{demos/perl/GMP.pm}.
@end itemize

As an aside, consideration has been given at various times to some sort of
expression evaluation within the main GMP library.  Going beyond something
minimal quickly leads to matters like user-defined functions, looping, fixnums
for control variables, etc, which are considered outside the scope of GMP
(much closer to language interpreters or compilers, @xref{Language Bindings}.)
Something simple for program input convenience may yet be a possibility, a
combination of the @file{expr} demo and the @file{pexpr} tree back-end
perhaps.  But for now the above evaluators are offered as illustrations.


@need 1000
@node Efficiency, Debugging, Demonstration Programs, GMP Basics
@section Efficiency
@cindex Efficiency

@table @asis
@item Small Operands
@cindex Small operands
On small operands, the time for function call overheads and memory allocation
can be significant in comparison to actual calculation.  This is unavoidable
in a general purpose variable precision library, although GMP attempts to be
as efficient as it can on both large and small operands.

@item Static Linking
@cindex Static linking
On some CPUs, in particular the x86s, the static @file{libgmp.a} should be
used for maximum speed, since the PIC code in the shared @file{libgmp.so} will
have a small overhead on each function call and global data address.  For many
programs this will be insignificant, but for long calculations there's a gain
to be had.

@item Initializing and Clearing
@cindex Initializing and clearing
Avoid excessive initializing and clearing of variables, since this can be
quite time consuming, especially in comparison to otherwise fast operations
like addition.

A language interpreter might want to keep a free list or stack of
initialized variables ready for use.  It should be possible to integrate
something like that with a garbage collector too.

@item Reallocations
@cindex Reallocations
An @code{mpz_t} or @code{mpq_t} variable used to hold successively increasing
values will have its memory repeatedly @code{realloc}ed, which could be quite
slow or could fragment memory, depending on the C library.  If an application
can estimate the final size then @code{mpz_init2} or @code{mpz_realloc2} can
be called to allocate the necessary space from the beginning
(@pxref{Initializing Integers}).

It doesn't matter if a size set with @code{mpz_init2} or @code{mpz_realloc2}
is too small, since all functions will do a further reallocation if necessary.
Badly overestimating memory required will waste space though.

@item @code{2exp} Functions
@cindex @code{2exp} functions
It's up to an application to call functions like @code{mpz_mul_2exp} when
appropriate.  General purpose functions like @code{mpz_mul} make no attempt to
identify powers of two or other special forms, because such inputs will
usually be very rare and testing every time would be wasteful.

@item @code{ui} and @code{si} Functions
@cindex @code{ui} and @code{si} functions
The @code{ui} functions and the small number of @code{si} functions exist for
convenience and should be used where applicable.  But if for example an
@code{mpz_t} contains a value that fits in an @code{unsigned long} there's no
need extract it and call a @code{ui} function, just use the regular @code{mpz}
function.

@item In-Place Operations
@cindex In-place operations
@code{mpz_abs}, @code{mpq_abs}, @code{mpf_abs}, @code{mpz_neg}, @code{mpq_neg}
and @code{mpf_neg} are fast when used for in-place operations like
@code{mpz_abs(x,x)}, since in the current implementation only a single field
of @code{x} needs changing.  On suitable compilers (GCC for instance) this is
inlined too.

@code{mpz_add_ui}, @code{mpz_sub_ui}, @code{mpf_add_ui} and @code{mpf_sub_ui}
benefit from an in-place operation like @code{mpz_add_ui(x,x,y)}, since
usually only one or two limbs of @code{x} will need to be changed.  The same
applies to the full precision @code{mpz_add} etc if @code{y} is small.  If
@code{y} is big then cache locality may be helped, but that's all.

@code{mpz_mul} is currently the opposite, a separate destination is slightly
better.  A call like @code{mpz_mul(x,x,y)} will, unless @code{y} is only one
limb, make a temporary copy of @code{x} before forming the result.  Normally
that copying will only be a tiny fraction of the time for the multiply, so
this is not a particularly important consideration.

@code{mpz_set}, @code{mpq_set}, @code{mpq_set_num}, @code{mpf_set}, etc, make
no attempt to recognise a copy of something to itself, so a call like
@code{mpz_set(x,x)} will be wasteful.  Naturally that would never be written
deliberately, but if it might arise from two pointers to the same object then
a test to avoid it might be desirable.

@example
if (x != y)
  mpz_set (x, y);
@end example

Note that it's never worth introducing extra @code{mpz_set} calls just to get
in-place operations.  If a result should go to a particular variable then just
direct it there and let GMP take care of data movement.

@item Divisibility Testing (Small Integers)
@cindex Divisibility testing
@code{mpz_divisible_ui_p} and @code{mpz_congruent_ui_p} are the best functions
for testing whether an @code{mpz_t} is divisible by an individual small
integer.  They use an algorithm which is faster than @code{mpz_tdiv_ui}, but
which gives no useful information about the actual remainder, only whether
it's zero (or a particular value).

However when testing divisibility by several small integers, it's best to take
a remainder modulo their product, to save multi-precision operations.  For
instance to test whether a number is divisible by any of 23, 29 or 31 take a
remainder modulo @math{23@times{}29@times{}31 = 20677} and then test that.

The division functions like @code{mpz_tdiv_q_ui} which give a quotient as well
as a remainder are generally a little slower than the remainder-only functions
like @code{mpz_tdiv_ui}.  If the quotient is only rarely wanted then it's
probably best to just take a remainder and then go back and calculate the
quotient if and when it's wanted (@code{mpz_divexact_ui} can be used if the
remainder is zero).

@item Rational Arithmetic
@cindex Rational arithmetic
The @code{mpq} functions operate on @code{mpq_t} values with no common factors
in the numerator and denominator.  Common factors are checked-for and cast out
as necessary.  In general, cancelling factors every time is the best approach
since it minimizes the sizes for subsequent operations.

However, applications that know something about the factorization of the
values they're working with might be able to avoid some of the GCDs used for
canonicalization, or swap them for divisions.  For example when multiplying by
a prime it's enough to check for factors of it in the denominator instead of
doing a full GCD@.  Or when forming a big product it might be known that very
little cancellation will be possible, and so canonicalization can be left to
the end.

The @code{mpq_numref} and @code{mpq_denref} macros give access to the
numerator and denominator to do things outside the scope of the supplied
@code{mpq} functions.  @xref{Applying Integer Functions}.

The canonical form for rationals allows mixed-type @code{mpq_t} and integer
additions or subtractions to be done directly with multiples of the
denominator.  This will be somewhat faster than @code{mpq_add}.  For example,

@example
/* mpq increment */
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));

/* mpq += unsigned long */
mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);

/* mpq -= mpz */
mpz_submul (mpq_numref(q), mpq_denref(q), z);
@end example

@item Number Sequences
@cindex Number sequences
Functions like @code{mpz_fac_ui}, @code{mpz_fib_ui} and @code{mpz_bin_uiui}
are designed for calculating isolated values.  If a range of values is wanted
it's probably best to get a starting point and iterate from there.

@item Text Input/Output
@cindex Text input/output
Hexadecimal or octal are suggested for input or output in text form.
Power-of-2 bases like these can be converted much more efficiently than other
bases, like decimal.  For big numbers there's usually nothing of particular
interest to be seen in the digits, so the base doesn't matter much.

Maybe we can hope octal will one day become the normal base for everyday use,
as proposed by King Charles XII of Sweden and later reformers.
@c Reference: Knuth volume 2 section 4.1, page 184 of second edition.  :-)
@end table


@node Debugging, Profiling, Efficiency, GMP Basics
@section Debugging
@cindex Debugging

@table @asis
@item Stack Overflow
@cindex Stack overflow
@cindex Segmentation violation
@cindex Bus error
Depending on the system, a segmentation violation or bus error might be the
only indication of stack overflow.  See @samp{--enable-alloca} choices in
@ref{Build Options}, for how to address this.

In new enough versions of GCC, @samp{-fstack-check} may be able to ensure an
overflow is recognised by the system before too much damage is done, or
@samp{-fstack-limit-symbol} or @samp{-fstack-limit-register} may be able to
add checking if the system itself doesn't do any (@pxref{Code Gen Options,,
Options for Code Generation, gcc, Using the GNU Compiler Collection (GCC)}).
These options must be added to the @samp{CFLAGS} used in the GMP build
(@pxref{Build Options}), adding them just to an application will have no
effect.  Note also they're a slowdown, adding overhead to each function call
and each stack allocation.

@item Heap Problems
@cindex Heap problems
@cindex Malloc problems
The most likely cause of application problems with GMP is heap corruption.
Failing to @code{init} GMP variables will have unpredictable effects, and
corruption arising elsewhere in a program may well affect GMP@.  Initializing
GMP variables more than once or failing to clear them will cause memory leaks.

@cindex Malloc debugger
In all such cases a @code{malloc} debugger is recommended.  On a GNU or BSD
system the standard C library @code{malloc} has some diagnostic facilities,
see @ref{Allocation Debugging,, Allocation Debugging, libc, The GNU C Library
Reference Manual}, or @samp{man 3 malloc}.  Other possibilities, in no
particular order, include

@display
@uref{http://cs.ecs.baylor.edu/~donahoo/tools/ccmalloc/}
@uref{http://dmalloc.com/}
@uref{https://wiki.gnome.org/Apps/MemProf}
@end display

The GMP default allocation routines in @file{memory.c} also have a simple
sentinel scheme which can be enabled with @code{#define DEBUG} in that file.
This is mainly designed for detecting buffer overruns during GMP development,
but might find other uses.

@item Stack Backtraces
@cindex Stack backtrace
On some systems the compiler options GMP uses by default can interfere with
debugging.  In particular on x86 and 68k systems @samp{-fomit-frame-pointer}
is used and this generally inhibits stack backtracing.  Recompiling without
such options may help while debugging, though the usual caveats about it
potentially moving a memory problem or hiding a compiler bug will apply.

@item GDB, the GNU Debugger
@cindex GDB
@cindex GNU Debugger
A sample @file{.gdbinit} is included in the distribution, showing how to call
some undocumented dump functions to print GMP variables from within GDB@.  Note
that these functions shouldn't be used in final application code since they're
undocumented and may be subject to incompatible changes in future versions of
GMP.

@item Source File Paths
GMP has multiple source files with the same name, in different directories.
For example @file{mpz}, @file{mpq} and @file{mpf} each have an
@file{init.c}.  If the debugger can't already determine the right one it may
help to build with absolute paths on each C file.  One way to do that is to
use a separate object directory with an absolute path to the source directory.

@example
cd /my/build/dir
/my/source/dir/gmp-@value{VERSION}/configure
@end example

This works via @code{VPATH}, and might require GNU @command{make}.
Alternately it might be possible to change the @code{.c.lo} rules
appropriately.

@item Assertion Checking
@cindex Assertion checking
The build option @option{--enable-assert} is available to add some consistency
checks to the library (see @ref{Build Options}).  These are likely to be of
limited value to most applications.  Assertion failures are just as likely to
indicate memory corruption as a library or compiler bug.

Applications using the low-level @code{mpn} functions, however, will benefit
from @option{--enable-assert} since it adds checks on the parameters of most
such functions, many of which have subtle restrictions on their usage.  Note
however that only the generic C code has checks, not the assembly code, so
@option{--disable-assembly} should be used for maximum checking.

@item Temporary Memory Checking
The build option @option{--enable-alloca=debug} arranges that each block of
temporary memory in GMP is allocated with a separate call to @code{malloc} (or
the allocation function set with @code{mp_set_memory_functions}).

This can help a malloc debugger detect accesses outside the intended bounds,
or detect memory not released.  In a normal build, on the other hand,
temporary memory is allocated in blocks which GMP divides up for its own use,
or may be allocated with a compiler builtin @code{alloca} which will go
nowhere near any malloc debugger hooks.

@item Maximum Debuggability
To summarize the above, a GMP build for maximum debuggability would be

@example
./configure --disable-shared --enable-assert \
  --enable-alloca=debug --disable-assembly CFLAGS=-g
@end example

For C++, add @samp{--enable-cxx CXXFLAGS=-g}.

@item Checker
@cindex Checker
@cindex GCC Checker
The GCC checker (@uref{https://savannah.nongnu.org/projects/checker/}) can be
used with GMP@.  It contains a stub library which means GMP applications
compiled with checker can use a normal GMP build.

A build of GMP with checking within GMP itself can be made.  This will run
very very slowly.  On GNU/Linux for example,

@cindex @command{checkergcc}
@example
./configure --disable-assembly CC=checkergcc
@end example

@option{--disable-assembly} must be used, since the GMP assembly code doesn't
support the checking scheme.  The GMP C++ features cannot be used, since
current versions of checker (0.9.9.1) don't yet support the standard C++
library.

@item Valgrind
@cindex Valgrind
Valgrind (@uref{http://valgrind.org/}) is a memory checker for x86, ARM, MIPS,
PowerPC, and S/390.  It translates and emulates machine instructions to do
strong checks for uninitialized data (at the level of individual bits), memory
accesses through bad pointers, and memory leaks.

Valgrind does not always support every possible instruction, in particular
ones recently added to an ISA.  Valgrind might therefore be incompatible with
a recent GMP or even a less recent GMP which is compiled using a recent GCC.

GMP's assembly code sometimes promotes a read of the limbs to some larger size,
for efficiency.  GMP will do this even at the start and end of a multilimb
operand, using naturally aligned operations on the larger type.  This may lead
to benign reads outside of allocated areas, triggering complaints from
Valgrind.  Valgrind's option @samp{--partial-loads-ok=yes} should help.

@item Other Problems
Any suspected bug in GMP itself should be isolated to make sure it's not an
application problem, see @ref{Reporting Bugs}.
@end table


@node Profiling, Autoconf, Debugging, GMP Basics
@section Profiling
@cindex Profiling
@cindex Execution profiling
@cindex @code{--enable-profiling}

Running a program under a profiler is a good way to find where it's spending
most time and where improvements can be best sought.  The profiling choices
for a GMP build are as follows.

@table @asis
@item @samp{--disable-profiling}
The default is to add nothing special for profiling.

It should be possible to just compile the mainline of a program with @code{-p}
and use @command{prof} to get a profile consisting of timer-based sampling of
the program counter.  Most of the GMP assembly code has the necessary symbol
information.

This approach has the advantage of minimizing interference with normal program
operation, but on most systems the resolution of the sampling is quite low (10
milliseconds for instance), requiring long runs to get accurate information.

@item @samp{--enable-profiling=prof}
@cindex @code{prof}
Build with support for the system @command{prof}, which means @samp{-p} added
to the @samp{CFLAGS}.

This provides call counting in addition to program counter sampling, which
allows the most frequently called routines to be identified, and an average
time spent in each routine to be determined.

The x86 assembly code has support for this option, but on other processors
the assembly routines will be as if compiled without @samp{-p} and therefore
won't appear in the call counts.

On some systems, such as GNU/Linux, @samp{-p} in fact means @samp{-pg} and in
this case @samp{--enable-profiling=gprof} described below should be used
instead.

@item @samp{--enable-profiling=gprof}
@cindex @code{gprof}
Build with support for @command{gprof}, which means @samp{-pg} added to the
@samp{CFLAGS}.

This provides call graph construction in addition to call counting and program
counter sampling, which makes it possible to count calls coming from different
locations.  For example the number of calls to @code{mpn_mul} from
@code{mpz_mul} versus the number from @code{mpf_mul}.  The program counter
sampling is still flat though, so only a total time in @code{mpn_mul} would be
accumulated, not a separate amount for each call site.

The x86 assembly code has support for this option, but on other processors
the assembly routines will be as if compiled without @samp{-pg} and therefore
not be included in the call counts.

On x86 and m68k systems @samp{-pg} and @samp{-fomit-frame-pointer} are
incompatible, so the latter is omitted from the default flags in that case,
which might result in poorer code generation.

Incidentally, it should be possible to use the @command{gprof} program with a
plain @samp{--enable-profiling=prof} build.  But in that case only the
@samp{gprof -p} flat profile and call counts can be expected to be valid, not
the @samp{gprof -q} call graph.

@item @samp{--enable-profiling=instrument}
@cindex @code{-finstrument-functions}
@cindex @code{instrument-functions}
Build with the GCC option @samp{-finstrument-functions} added to the
@samp{CFLAGS} (@pxref{Code Gen Options,, Options for Code Generation, gcc,
Using the GNU Compiler Collection (GCC)}).

This inserts special instrumenting calls at the start and end of each
function, allowing exact timing and full call graph construction.

This instrumenting is not normally a standard system feature and will require
support from an external library, such as

@cindex FunctionCheck
@cindex fnccheck
@display
@uref{https://sourceforge.net/projects/fnccheck/}
@end display

This should be included in @samp{LIBS} during the GMP configure so that test
programs will link.  For example,

@example
./configure --enable-profiling=instrument LIBS=-lfc
@end example

On a GNU system the C library provides dummy instrumenting functions, so
programs compiled with this option will link.  In this case it's only
necessary to ensure the correct library is added when linking an application.

The x86 assembly code supports this option, but on other processors the
assembly routines will be as if compiled without
@samp{-finstrument-functions} meaning time spent in them will effectively be
attributed to their caller.
@end table


@node Autoconf, Emacs, Profiling, GMP Basics
@section Autoconf
@cindex Autoconf

Autoconf based applications can easily check whether GMP is installed.  The
only thing to be noted is that GMP library symbols from version 3 onwards have
prefixes like @code{__gmpz}.  The following therefore would be a simple test,

@cindex @code{AC_CHECK_LIB}
@example
AC_CHECK_LIB(gmp, __gmpz_init)
@end example

This just uses the default @code{AC_CHECK_LIB} actions for found or not found,
but an application that must have GMP would want to generate an error if not
found.  For example,

@example
AC_CHECK_LIB(gmp, __gmpz_init, ,
  [AC_MSG_ERROR([GNU MP not found, see https://gmplib.org/])])
@end example

If functions added in some particular version of GMP are required, then one of
those can be used when checking.  For example @code{mpz_mul_si} was added in
GMP 3.1,

@example
AC_CHECK_LIB(gmp, __gmpz_mul_si, ,
  [AC_MSG_ERROR(
  [GNU MP not found, or not 3.1 or up, see https://gmplib.org/])])
@end example

An alternative would be to test the version number in @file{gmp.h} using say
@code{AC_EGREP_CPP}.  That would make it possible to test the exact version,
if some particular sub-minor release is known to be necessary.

In general it's recommended that applications should simply demand a new
enough GMP rather than trying to provide supplements for features not
available in past versions.

Occasionally an application will need or want to know the size of a type at
configuration or preprocessing time, not just with @code{sizeof} in the code.
This can be done in the normal way with @code{mp_limb_t} etc, but GMP 4.0 or
up is best for this, since prior versions needed certain @samp{-D} defines on
systems using a @code{long long} limb.  The following would suit Autoconf 2.50
or up,

@example
AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])
@end example


@node Emacs,  , Autoconf, GMP Basics
@section Emacs
@cindex Emacs
@cindex @code{info-lookup-symbol}

@key{C-h C-i} (@code{info-lookup-symbol}) is a good way to find documentation
on C functions while editing (@pxref{Info Lookup, , Info Documentation Lookup,
emacs, The Emacs Editor}).

The GMP manual can be included in such lookups by putting the following in
your @file{.emacs},

@c  This isn't pretty, but there doesn't seem to be a better way (in emacs
@c  21.2 at least).  info-lookup->mode-value could be used for the "assoc"s,
@c  but that function isn't documented, whereas info-lookup-alist is.
@c
@example
(eval-after-load "info-look"
  '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
     (setcar (nthcdr 3 mode-value)
             (cons '("(gmp)Function Index" nil "^ -.* " "\\>")
                   (nth 3 mode-value)))))
@end example


@node Reporting Bugs, Integer Functions, GMP Basics, Top
@comment  node-name,  next,  previous,  up
@chapter Reporting Bugs
@cindex Reporting bugs
@cindex Bug reporting

If you think you have found a bug in the GMP library, please investigate it
and report it.  We have made this library available to you, and it is not too
much to ask you to report the bugs you find.

Before you report a bug, check it's not already addressed in @ref{Known Build
Problems}, or perhaps @ref{Notes for Particular Systems}.  You may also want
to check @uref{https://gmplib.org/} for patches for this release.

Please include the following in any report,

@itemize @bullet
@item
The GMP version number, and if pre-packaged or patched then say so.

@item
A test program that makes it possible for us to reproduce the bug.  Include
instructions on how to run the program.

@item
A description of what is wrong.  If the results are incorrect, in what way.
If you get a crash, say so.

@item
If you get a crash, include a stack backtrace from the debugger if it's
informative (@samp{where} in @command{gdb}, or @samp{$C} in @command{adb}).

@item
Please do not send core dumps, executables or @command{strace}s.

@item
The @samp{configure} options you used when building GMP, if any.

@item
The output from @samp{configure}, as printed to stdout, with any options used.

@item
The name of the compiler and its version.  For @command{gcc}, get the version
with @samp{gcc -v}, otherwise perhaps @samp{what `which cc`}, or similar.

@item
The output from running @samp{uname -a}.

@item
The output from running @samp{./config.guess}, and from running
@samp{./configfsf.guess} (might be the same).

@item
If the bug is related to @samp{configure}, then the compressed contents of
@file{config.log}.

@item
If the bug is related to an @file{asm} file not assembling, then the contents
of @file{config.m4} and the offending line or lines from the temporary
@file{mpn/tmp-<file>.s}.
@end itemize

Please make an effort to produce a self-contained report, with something
definite that can be tested or debugged.  Vague queries or piecemeal messages
are difficult to act on and don't help the development effort.

It is not uncommon that an observed problem is actually due to a bug in the
compiler; the GMP code tends to explore interesting corners in compilers.

If your bug report is good, we will do our best to help you get a corrected
version of the library; if the bug report is poor, we won't do anything about
it (except maybe ask you to send a better report).

Send your report to: @email{gmp-bugs@@gmplib.org}.

If you think something in this manual is unclear, or downright incorrect, or if
the language needs to be improved, please send a note to the same address.


@node Integer Functions, Rational Number Functions, Reporting Bugs, Top
@comment  node-name,  next,  previous,  up
@chapter Integer Functions
@cindex Integer functions

This chapter describes the GMP functions for performing integer arithmetic.
These functions start with the prefix @code{mpz_}.

GMP integers are stored in objects of type @code{mpz_t}.

@menu
* Initializing Integers::
* Assigning Integers::
* Simultaneous Integer Init & Assign::
* Converting Integers::
* Integer Arithmetic::
* Integer Division::
* Integer Exponentiation::
* Integer Roots::
* Number Theoretic Functions::
* Integer Comparisons::
* Integer Logic and Bit Fiddling::
* I/O of Integers::
* Integer Random Numbers::
* Integer Import and Export::
* Miscellaneous Integer Functions::
* Integer Special Functions::
@end menu

@node Initializing Integers, Assigning Integers, Integer Functions, Integer Functions
@comment  node-name,  next,  previous,  up
@section Initialization Functions
@cindex Integer initialization functions
@cindex Initialization functions

The functions for integer arithmetic assume that all integer objects are
initialized.  You do that by calling the function @code{mpz_init}.  For
example,

@example
@{
  mpz_t integ;
  mpz_init (integ);
  @dots{}
  mpz_add (integ, @dots{});
  @dots{}
  mpz_sub (integ, @dots{});

  /* Unless the program is about to exit, do ... */
  mpz_clear (integ);
@}
@end example

As you can see, you can store new values any number of times, once an
object is initialized.

@deftypefun void mpz_init (mpz_t @var{x})
Initialize @var{x}, and set its value to 0.
@end deftypefun

@deftypefun void mpz_inits (mpz_t @var{x}, ...)
Initialize a NULL-terminated list of @code{mpz_t} variables, and set their
values to 0.
@end deftypefun

@deftypefun void mpz_init2 (mpz_t @var{x}, mp_bitcnt_t @var{n})
Initialize @var{x}, with space for @var{n}-bit numbers, and set its value to 0.
Calling this function instead of @code{mpz_init} or @code{mpz_inits} is never
necessary; reallocation is handled automatically by GMP when needed.

While @var{n} defines the initial space, @var{x} will grow automatically in the
normal way, if necessary, for subsequent values stored.  @code{mpz_init2} makes
it possible to avoid such reallocations if a maximum size is known in advance.

In preparation for an operation, GMP often allocates one limb more than
ultimately needed.  To make sure GMP will not perform reallocation for
@var{x}, you need to add the number of bits in @code{mp_limb_t} to @var{n}.
@end deftypefun

@deftypefun void mpz_clear (mpz_t @var{x})
Free the space occupied by @var{x}.  Call this function for all @code{mpz_t}
variables when you are done with them.
@end deftypefun

@deftypefun void mpz_clears (mpz_t @var{x}, ...)
Free the space occupied by a NULL-terminated list of @code{mpz_t} variables.
@end deftypefun

@deftypefun void mpz_realloc2 (mpz_t @var{x}, mp_bitcnt_t @var{n})
Change the space allocated for @var{x} to @var{n} bits.  The value in @var{x}
is preserved if it fits, or is set to 0 if not.

Calling this function is never necessary; reallocation is handled automatically
by GMP when needed.  But this function can be used to increase the space for a
variable in order to avoid repeated automatic reallocations, or to decrease it
to give memory back to the heap.
@end deftypefun


@node Assigning Integers, Simultaneous Integer Init & Assign, Initializing Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Assignment Functions
@cindex Integer assignment functions
@cindex Assignment functions

These functions assign new values to already initialized integers
(@pxref{Initializing Integers}).

@deftypefun void mpz_set (mpz_t @var{rop}, const mpz_t @var{op})
@deftypefunx void mpz_set_ui (mpz_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpz_set_si (mpz_t @var{rop}, signed long int @var{op})
@deftypefunx void mpz_set_d (mpz_t @var{rop}, double @var{op})
@deftypefunx void mpz_set_q (mpz_t @var{rop}, const mpq_t @var{op})
@deftypefunx void mpz_set_f (mpz_t @var{rop}, const mpf_t @var{op})
Set the value of @var{rop} from @var{op}.

@code{mpz_set_d}, @code{mpz_set_q} and @code{mpz_set_f} truncate @var{op} to
make it an integer.
@end deftypefun

@deftypefun int mpz_set_str (mpz_t @var{rop}, const char *@var{str}, int @var{base})
Set the value of @var{rop} from @var{str}, a null-terminated C string in base
@var{base}.  White space is allowed in the string, and is simply ignored.

The @var{base} may vary from 2 to 62, or if @var{base} is 0, then the leading
characters are used: @code{0x} and @code{0X} for hexadecimal, @code{0b} and
@code{0B} for binary, @code{0} for octal, or decimal otherwise.

For bases up to 36, case is ignored; upper-case and lower-case letters have
the same value.  For bases 37 to 62, upper-case letter represent the usual
10..35 while lower-case letter represent 36..61.

This function returns 0 if the entire string is a valid number in base
@var{base}.  Otherwise it returns @minus{}1.
@c
@c  It turns out that it is not entirely true that this function ignores
@c  white-space.  It does ignore it between digits, but not after a minus sign
@c  or within or after ``0x''.  Some thought was given to disallowing all
@c  whitespace, but that would be an incompatible change, whitespace has been
@c  documented as ignored ever since GMP 1.
@c
@end deftypefun

@deftypefun void mpz_swap (mpz_t @var{rop1}, mpz_t @var{rop2})
Swap the values @var{rop1} and @var{rop2} efficiently.
@end deftypefun


@node Simultaneous Integer Init & Assign, Converting Integers, Assigning Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Combined Initialization and Assignment Functions
@cindex Integer assignment functions
@cindex Assignment functions
@cindex Integer initialization functions
@cindex Initialization functions

For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there.  These functions'
names have the form @code{mpz_init_set@dots{}}

Here is an example of using one:

@example
@{
  mpz_t pie;
  mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
  @dots{}
  mpz_sub (pie, @dots{});
  @dots{}
  mpz_clear (pie);
@}
@end example

@noindent
Once the integer has been initialized by any of the @code{mpz_init_set@dots{}}
functions, it can be used as the source or destination operand for the ordinary
integer functions.  Don't use an initialize-and-set function on a variable
already initialized!

@deftypefun void mpz_init_set (mpz_t @var{rop}, const mpz_t @var{op})
@deftypefunx void mpz_init_set_ui (mpz_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpz_init_set_si (mpz_t @var{rop}, signed long int @var{op})
@deftypefunx void mpz_init_set_d (mpz_t @var{rop}, double @var{op})
Initialize @var{rop} with limb space and set the initial numeric value from
@var{op}.
@end deftypefun

@deftypefun int mpz_init_set_str (mpz_t @var{rop}, const char *@var{str}, int @var{base})
Initialize @var{rop} and set its value like @code{mpz_set_str} (see its
documentation above for details).

If the string is a correct base @var{base} number, the function returns 0;
if an error occurs it returns @minus{}1.  @var{rop} is initialized even if
an error occurs.  (I.e., you have to call @code{mpz_clear} for it.)
@end deftypefun


@node Converting Integers, Integer Arithmetic, Simultaneous Integer Init & Assign, Integer Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Integer conversion functions
@cindex Conversion functions

This section describes functions for converting GMP integers to standard C
types.  Functions for converting @emph{to} GMP integers are described in
@ref{Assigning Integers} and @ref{I/O of Integers}.

@deftypefun {unsigned long int} mpz_get_ui (const mpz_t @var{op})
Return the value of @var{op} as an @code{unsigned long}.

If @var{op} is too big to fit an @code{unsigned long} then just the least
significant bits that do fit are returned.  The sign of @var{op} is ignored,
only the absolute value is used.
@end deftypefun

@deftypefun {signed long int} mpz_get_si (const mpz_t @var{op})
If @var{op} fits into a @code{signed long int} return the value of @var{op}.
Otherwise return the least significant part of @var{op}, with the same sign
as @var{op}.

If @var{op} is too big to fit in a @code{signed long int}, the returned
result is probably not very useful.  To find out if the value will fit, use
the function @code{mpz_fits_slong_p}.
@end deftypefun

@deftypefun double mpz_get_d (const mpz_t @var{op})
Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding
towards zero).

If the exponent from the conversion is too big, the result is system
dependent.  An infinity is returned where available.  A hardware overflow trap
may or may not occur.
@end deftypefun

@deftypefun double mpz_get_d_2exp (signed long int *@var{exp}, const mpz_t @var{op})
Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding
towards zero), and returning the exponent separately.

The return value is in the range @math{0.5@le{}@GMPabs{@var{d}}<1} and the
exponent is stored to @code{*@var{exp}}.  @m{@var{d} * 2^{exp}, @var{d} *
2^@var{exp}} is the (truncated) @var{op} value.  If @var{op} is zero, the
return is @math{0.0} and 0 is stored to @code{*@var{exp}}.

@cindex @code{frexp}
This is similar to the standard C @code{frexp} function (@pxref{Normalization
Functions,,, libc, The GNU C Library Reference Manual}).
@end deftypefun

@deftypefun {char *} mpz_get_str (char *@var{str}, int @var{base}, const mpz_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  The base argument
may vary from 2 to 62 or from @minus{}2 to @minus{}36.

For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of storage large
enough for the result, that being @code{mpz_sizeinbase (@var{op}, @var{base})
+ 2}.  The two extra bytes are for a possible minus sign, and the
null-terminator.

A pointer to the result string is returned, being either the allocated block,
or the given @var{str}.
@end deftypefun


@need 2000
@node Integer Arithmetic, Integer Division, Converting Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Integer arithmetic functions
@cindex Arithmetic functions

@deftypefun void mpz_add (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
@deftypefunx void mpz_add_ui (mpz_t @var{rop}, const mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{op1} + @var{op2}}.
@end deftypefun

@deftypefun void mpz_sub (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
@deftypefunx void mpz_sub_ui (mpz_t @var{rop}, const mpz_t @var{op1}, unsigned long int @var{op2})
@deftypefunx void mpz_ui_sub (mpz_t @var{rop}, unsigned long int @var{op1}, const mpz_t @var{op2})
Set @var{rop} to @var{op1} @minus{} @var{op2}.
@end deftypefun

@deftypefun void mpz_mul (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
@deftypefunx void mpz_mul_si (mpz_t @var{rop}, const mpz_t @var{op1}, long int @var{op2})
@deftypefunx void mpz_mul_ui (mpz_t @var{rop}, const mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_addmul (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
@deftypefunx void mpz_addmul_ui (mpz_t @var{rop}, const mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{rop} + @var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_submul (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
@deftypefunx void mpz_submul_ui (mpz_t @var{rop}, const mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{rop} - @var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_mul_2exp (mpz_t @var{rop}, const mpz_t @var{op1}, mp_bitcnt_t @var{op2})
@cindex Bit shift left
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.  This operation can also be defined as a left shift by @var{op2}
bits.
@end deftypefun

@deftypefun void mpz_neg (mpz_t @var{rop}, const mpz_t @var{op})
Set @var{rop} to @minus{}@var{op}.
@end deftypefun

@deftypefun void mpz_abs (mpz_t @var{rop}, const mpz_t @var{op})
Set @var{rop} to the absolute value of @var{op}.
@end deftypefun


@need 2000
@node Integer Division, Integer Exponentiation, Integer Arithmetic, Integer Functions
@section Division Functions
@cindex Integer division functions
@cindex Division functions

Division is undefined if the divisor is zero.  Passing a zero divisor to the
division or modulo functions (including the modular powering functions
@code{mpz_powm} and @code{mpz_powm_ui}), will cause an intentional division by
zero.  This lets a program handle arithmetic exceptions in these functions the
same way as for normal C @code{int} arithmetic.

@c  Separate deftypefun groups for cdiv, fdiv and tdiv produce a blank line
@c  between each, and seem to let tex do a better job of page breaks than an
@c  @sp 1 in the middle of one big set.

@deftypefun void mpz_cdiv_q (mpz_t @var{q}, const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx void mpz_cdiv_r (mpz_t @var{r}, const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, const mpz_t @var{n}, const mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_cdiv_q_ui (mpz_t @var{q}, const mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{r}, const mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{const mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_ui (const mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_cdiv_q_2exp (mpz_t @var{q}, const mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
@deftypefunx void mpz_cdiv_r_2exp (mpz_t @var{r}, const mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
@end deftypefun

@deftypefun void mpz_fdiv_q (mpz_t @var{q}, const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx void mpz_fdiv_r (mpz_t @var{r}, const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx void mpz_fdiv_qr (mpz_t @var{q}, mpz_t @var{r}, const mpz_t @var{n}, const mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_fdiv_q_ui (mpz_t @var{q}, const mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_r_ui (mpz_t @var{r}, const mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{const mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_ui (const mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_fdiv_q_2exp (mpz_t @var{q}, const mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
@deftypefunx void mpz_fdiv_r_2exp (mpz_t @var{r}, const mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
@end deftypefun

@deftypefun void mpz_tdiv_q (mpz_t @var{q}, const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx void mpz_tdiv_r (mpz_t @var{r}, const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, const mpz_t @var{n}, const mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_tdiv_q_ui (mpz_t @var{q}, const mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_r_ui (mpz_t @var{r}, const mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{const mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_ui (const mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_tdiv_q_2exp (mpz_t @var{q}, const mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
@deftypefunx void mpz_tdiv_r_2exp (mpz_t @var{r}, const mpz_t @var{n}, @w{mp_bitcnt_t @var{b}})
@cindex Bit shift right

@sp 1
Divide @var{n} by @var{d}, forming a quotient @var{q} and/or remainder
@var{r}.  For the @code{2exp} functions, @m{@var{d}=2^b, @var{d}=2^@var{b}}.
The rounding is in three styles, each suiting different applications.

@itemize @bullet
@item
@code{cdiv} rounds @var{q} up towards @m{+\infty, +infinity}, and @var{r} will
have the opposite sign to @var{d}.  The @code{c} stands for ``ceil''.

@item
@code{fdiv} rounds @var{q} down towards @m{-\infty, @minus{}infinity}, and
@var{r} will have the same sign as @var{d}.  The @code{f} stands for
``floor''.

@item
@code{tdiv} rounds @var{q} towards zero, and @var{r} will have the same sign
as @var{n}.  The @code{t} stands for ``truncate''.
@end itemize

In all cases @var{q} and @var{r} will satisfy
@m{@var{n}=@var{q}@var{d}+@var{r}, @var{n}=@var{q}*@var{d}+@var{r}}, and
@var{r} will satisfy @math{0@le{}@GMPabs{@var{r}}<@GMPabs{@var{d}}}.

The @code{q} functions calculate only the quotient, the @code{r} functions
only the remainder, and the @code{qr} functions calculate both.  Note that for
@code{qr} the same variable cannot be passed for both @var{q} and @var{r}, or
results will be unpredictable.

For the @code{ui} variants the return value is the remainder, and in fact
returning the remainder is all the @code{div_ui} functions do.  For
@code{tdiv} and @code{cdiv} the remainder can be negative, so for those the
return value is the absolute value of the remainder.

For the @code{2exp} variants the divisor is @m{2^b,2^@var{b}}.  These
functions are implemented as right shifts and bit masks, but of course they
round the same as the other functions.

For positive @var{n} both @code{mpz_fdiv_q_2exp} and @code{mpz_tdiv_q_2exp}
are simple bitwise right shifts.  For negative @var{n}, @code{mpz_fdiv_q_2exp}
is effectively an arithmetic right shift treating @var{n} as twos complement
the same as the bitwise logical functions do, whereas @code{mpz_tdiv_q_2exp}
effectively treats @var{n} as sign and magnitude.
@end deftypefun

@deftypefun void mpz_mod (mpz_t @var{r}, const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_mod_ui (mpz_t @var{r}, const mpz_t @var{n}, @w{unsigned long int @var{d}})
Set @var{r} to @var{n} @code{mod} @var{d}.  The sign of the divisor is
ignored; the result is always non-negative.

@code{mpz_mod_ui} is identical to @code{mpz_fdiv_r_ui} above, returning the
remainder as well as setting @var{r}.  See @code{mpz_fdiv_ui} above if only
the return value is wanted.
@end deftypefun

@deftypefun void mpz_divexact (mpz_t @var{q}, const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx void mpz_divexact_ui (mpz_t @var{q}, const mpz_t @var{n}, unsigned long @var{d})
@cindex Exact division functions
Set @var{q} to @var{n}/@var{d}.  These functions produce correct results only
when it is known in advance that @var{d} divides @var{n}.

These routines are much faster than the other division functions, and are the
best choice when exact division is known to occur, for example reducing a
rational to lowest terms.
@end deftypefun

@deftypefun int mpz_divisible_p (const mpz_t @var{n}, const mpz_t @var{d})
@deftypefunx int mpz_divisible_ui_p (const mpz_t @var{n}, unsigned long int @var{d})
@deftypefunx int mpz_divisible_2exp_p (const mpz_t @var{n}, mp_bitcnt_t @var{b})
@cindex Divisibility functions
Return non-zero if @var{n} is exactly divisible by @var{d}, or in the case of
@code{mpz_divisible_2exp_p} by @m{2^b,2^@var{b}}.

@var{n} is divisible by @var{d} if there exists an integer @var{q} satisfying
@math{@var{n} = @var{q}@GMPmultiply{}@var{d}}.  Unlike the other division
functions, @math{@var{d}=0} is accepted and following the rule it can be seen
that only 0 is considered divisible by 0.
@end deftypefun

@deftypefun int mpz_congruent_p (const mpz_t @var{n}, const mpz_t @var{c}, const mpz_t @var{d})
@deftypefunx int mpz_congruent_ui_p (const mpz_t @var{n}, unsigned long int @var{c}, unsigned long int @var{d})
@deftypefunx int mpz_congruent_2exp_p (const mpz_t @var{n}, const mpz_t @var{c}, mp_bitcnt_t @var{b})
@cindex Divisibility functions
@cindex Congruence functions
Return non-zero if @var{n} is congruent to @var{c} modulo @var{d}, or in the
case of @code{mpz_congruent_2exp_p} modulo @m{2^b,2^@var{b}}.

@var{n} is congruent to @var{c} mod @var{d} if there exists an integer @var{q}
satisfying @math{@var{n} = @var{c} + @var{q}@GMPmultiply{}@var{d}}.  Unlike
the other division functions, @math{@var{d}=0} is accepted and following the
rule it can be seen that @var{n} and @var{c} are considered congruent mod 0
only when exactly equal.
@end deftypefun


@need 2000
@node Integer Exponentiation, Integer Roots, Integer Division, Integer Functions
@section Exponentiation Functions
@cindex Integer exponentiation functions
@cindex Exponentiation functions
@cindex Powering functions

@deftypefun void mpz_powm (mpz_t @var{rop}, const mpz_t @var{base}, const mpz_t @var{exp}, const mpz_t @var{mod})
@deftypefunx void mpz_powm_ui (mpz_t @var{rop}, const mpz_t @var{base}, unsigned long int @var{exp}, const mpz_t @var{mod})
Set @var{rop} to @m{base^{exp} \bmod mod, (@var{base} raised to @var{exp})
modulo @var{mod}}.

Negative @var{exp} is supported if the inverse @mm{@var{base}@sup{-1} @bmod
@var{mod}, @var{base}^(-1) @bmod @var{mod}} exists (see @code{mpz_invert} in
@ref{Number Theoretic Functions}).  If an inverse doesn't exist then a divide
by zero is raised.
@end deftypefun

@deftypefun void mpz_powm_sec (mpz_t @var{rop}, const mpz_t @var{base}, const mpz_t @var{exp}, const mpz_t @var{mod})
Set @var{rop} to @m{base^{exp} \bmod @var{mod}, (@var{base} raised to @var{exp})
modulo @var{mod}}.

It is required that @math{@var{exp} > 0} and that @var{mod} is odd.

This function is designed to take the same time and have the same cache access
patterns for any two same-size arguments, assuming that function arguments are
placed at the same position and that the machine state is identical upon
function entry.  This function is intended for cryptographic purposes, where
resilience to side-channel attacks is desired.
@end deftypefun

@deftypefun void mpz_pow_ui (mpz_t @var{rop}, const mpz_t @var{base}, unsigned long int @var{exp})
@deftypefunx void mpz_ui_pow_ui (mpz_t @var{rop}, unsigned long int @var{base}, unsigned long int @var{exp})
Set @var{rop} to @m{base^{exp}, @var{base} raised to @var{exp}}.  The case
@math{0^0} yields 1.
@end deftypefun


@need 2000
@node Integer Roots, Number Theoretic Functions, Integer Exponentiation, Integer Functions
@section Root Extraction Functions
@cindex Integer root functions
@cindex Root extraction functions

@deftypefun int mpz_root (mpz_t @var{rop}, const mpz_t @var{op}, unsigned long int @var{n})
Set @var{rop} to @m{\lfloor\root n \of {op}\rfloor@C{},} the truncated integer
part of the @var{n}th root of @var{op}.  Return non-zero if the computation
was exact, i.e., if @var{op} is @var{rop} to the @var{n}th power.
@end deftypefun

@deftypefun void mpz_rootrem (mpz_t @var{root}, mpz_t @var{rem}, const mpz_t @var{u}, unsigned long int @var{n})
Set @var{root} to @m{\lfloor\root n \of {u}\rfloor@C{},} the truncated
integer part of the @var{n}th root of @var{u}.  Set @var{rem} to the
remainder, @m{(@var{u} - @var{root}^n),
@var{u}@minus{}@var{root}**@var{n}}.
@end deftypefun

@deftypefun void mpz_sqrt (mpz_t @var{rop}, const mpz_t @var{op})
Set @var{rop} to @m{\lfloor\sqrt{@var{op}}\rfloor@C{},} the truncated
integer part of the square root of @var{op}.
@end deftypefun

@deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, const mpz_t @var{op})
Set @var{rop1} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
of the square root of @var{op}}, like @code{mpz_sqrt}.  Set @var{rop2} to the
remainder @m{(@var{op} - @var{rop1}^2),
@var{op}@minus{}@var{rop1}*@var{rop1}}, which will be zero if @var{op} is a
perfect square.

If @var{rop1} and @var{rop2} are the same variable, the results are
undefined.
@end deftypefun

@deftypefun int mpz_perfect_power_p (const mpz_t @var{op})
@cindex Perfect power functions
@cindex Root testing functions
Return non-zero if @var{op} is a perfect power, i.e., if there exist integers
@m{a,@var{a}} and @m{b,@var{b}}, with @m{b>1, @var{b}>1}, such that
@m{@var{op}=a^b, @var{op} equals @var{a} raised to the power @var{b}}.

Under this definition both 0 and 1 are considered to be perfect powers.
Negative values of @var{op} are accepted, but of course can only be odd
perfect powers.
@end deftypefun

@deftypefun int mpz_perfect_square_p (const mpz_t @var{op})
@cindex Perfect square functions
@cindex Root testing functions
Return non-zero if @var{op} is a perfect square, i.e., if the square root of
@var{op} is an integer.  Under this definition both 0 and 1 are considered to
be perfect squares.
@end deftypefun


@need 2000
@node Number Theoretic Functions, Integer Comparisons, Integer Roots, Integer Functions
@section Number Theoretic Functions
@cindex Number theoretic functions

@deftypefun int mpz_probab_prime_p (const mpz_t @var{n}, int @var{reps})
@cindex Prime testing functions
@cindex Probable prime testing functions
Determine whether @var{n} is prime.  Return 2 if @var{n} is definitely prime,
return 1 if @var{n} is probably prime (without being certain), or return 0 if
@var{n} is definitely non-prime.

This function performs some trial divisions, a Baillie-PSW probable prime
test, then @var{reps-24} Miller-Rabin probabilistic primality tests.  A
higher @var{reps} value will reduce the chances of a non-prime being
identified as ``probably prime''.  A composite number will be identified as a
prime with an asymptotic probability of less than @m{4^{-reps},4^(-@var{reps})}.
Reasonable values of @var{reps} are between 15 and 50.

GMP versions up to and including 6.1.2 did not use the Baillie-PSW
primality test. In those older versions of GMP, this function performed
@var{reps} Miller-Rabin tests.
@end deftypefun

@deftypefun void mpz_nextprime (mpz_t @var{rop}, const mpz_t @var{op})
@cindex Next prime function
Set @var{rop} to the next prime greater than @var{op}.

This function uses a probabilistic algorithm to identify primes.  For
practical purposes it's adequate, the chance of a composite passing will be
extremely small.
@end deftypefun

@c mpz_prime_p not implemented as of gmp 3.0.

@c @deftypefun int mpz_prime_p (const mpz_t @var{n})
@c Return non-zero if @var{n} is prime and zero if @var{n} is a non-prime.
@c This function is far slower than @code{mpz_probab_prime_p}, but then it
@c never returns non-zero for composite numbers.

@c (For practical purposes, using @code{mpz_probab_prime_p} is adequate.
@c The likelihood of a programming error or hardware malfunction is orders
@c of magnitudes greater than the likelihood for a composite to pass as a
@c prime, if the @var{reps} argument is in the suggested range.)
@c @end deftypefun

@deftypefun void mpz_gcd (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
@cindex Greatest common divisor functions
@cindex GCD functions
Set @var{rop} to the greatest common divisor of @var{op1} and @var{op2}.  The
result is always positive even if one or both input operands are negative.
Except if both inputs are zero; then this function defines @math{gcd(0,0) = 0}.
@end deftypefun

@deftypefun {unsigned long int} mpz_gcd_ui (mpz_t @var{rop}, const mpz_t @var{op1}, unsigned long int @var{op2})
Compute the greatest common divisor of @var{op1} and @var{op2}.  If
@var{rop} is not @code{NULL}, store the result there.

If the result is small enough to fit in an @code{unsigned long int}, it is
returned.  If the result does not fit, 0 is returned, and the result is equal
to the argument @var{op1}.  Note that the result will always fit if @var{op2}
is non-zero.
@end deftypefun

@deftypefun void mpz_gcdext (mpz_t @var{g}, mpz_t @var{s}, mpz_t @var{t}, const mpz_t @var{a}, const mpz_t @var{b})
@cindex Extended GCD
@cindex GCD extended
Set @var{g} to the greatest common divisor of @var{a} and @var{b}, and in
addition set @var{s} and @var{t} to coefficients satisfying
@math{@var{a}@GMPmultiply{}@var{s} + @var{b}@GMPmultiply{}@var{t} = @var{g}}.
The value in @var{g} is always positive, even if one or both of @var{a} and
@var{b} are negative (or zero if both inputs are zero).  The values in @var{s}
and @var{t} are chosen such that normally, @math{@GMPabs{@var{s}} <
@GMPabs{@var{b}} / (2 @var{g})} and @math{@GMPabs{@var{t}} < @GMPabs{@var{a}}
/ (2 @var{g})}, and these relations define @var{s} and @var{t} uniquely.  There
are a few exceptional cases:

If @math{@GMPabs{@var{a}} = @GMPabs{@var{b}}}, then @math{@var{s} = 0},
@math{@var{t} = sgn(@var{b})}.

Otherwise, @math{@var{s} = sgn(@var{a})} if @math{@var{b} = 0} or
@math{@GMPabs{@var{b}} = 2 @var{g}}, and @math{@var{t} = sgn(@var{b})} if
@math{@var{a} = 0} or @math{@GMPabs{@var{a}} = 2 @var{g}}.

In all cases, @math{@var{s} = 0} if and only if @math{@var{g} =
@GMPabs{@var{b}}}, i.e., if @var{b} divides @var{a} or @math{@var{a} = @var{b}
= 0}.

If @var{t} or @var{g} is @code{NULL} then that value is not computed.
@end deftypefun

@deftypefun void mpz_lcm (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
@deftypefunx void mpz_lcm_ui (mpz_t @var{rop}, const mpz_t @var{op1}, unsigned long @var{op2})
@cindex Least common multiple functions
@cindex LCM functions
Set @var{rop} to the least common multiple of @var{op1} and @var{op2}.
@var{rop} is always positive, irrespective of the signs of @var{op1} and
@var{op2}.  @var{rop} will be zero if either @var{op1} or @var{op2} is zero.
@end deftypefun

@deftypefun int mpz_invert (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
@cindex Modular inverse functions
@cindex Inverse modulo functions
Compute the inverse of @var{op1} modulo @var{op2} and put the result in
@var{rop}.  If the inverse exists, the return value is non-zero and @var{rop}
will satisfy @math{0 @le{} @var{rop} < @GMPabs{@var{op2}}} (with @math{@var{rop}
= 0} possible only when @math{@GMPabs{@var{op2}} = 1}, i.e., in the
somewhat degenerate zero ring).  If an inverse doesn't
exist the return value is zero and @var{rop} is undefined.  The behaviour of
this function is undefined when @var{op2} is zero.
@end deftypefun

@deftypefun int mpz_jacobi (const mpz_t @var{a}, const mpz_t @var{b})
@cindex Jacobi symbol functions
Calculate the Jacobi symbol @m{\left(a \over b\right),
(@var{a}/@var{b})}.  This is defined only for @var{b} odd.
@end deftypefun

@deftypefun int mpz_legendre (const mpz_t @var{a}, const mpz_t @var{p})
@cindex Legendre symbol functions
Calculate the Legendre symbol @m{\left(a \over p\right),
(@var{a}/@var{p})}.  This is defined only for @var{p} an odd positive
prime, and for such @var{p} it's identical to the Jacobi symbol.
@end deftypefun

@deftypefun int mpz_kronecker (const mpz_t @var{a}, const mpz_t @var{b})
@deftypefunx int mpz_kronecker_si (const mpz_t @var{a}, long @var{b})
@deftypefunx int mpz_kronecker_ui (const mpz_t @var{a}, unsigned long @var{b})
@deftypefunx int mpz_si_kronecker (long @var{a}, const mpz_t @var{b})
@deftypefunx int mpz_ui_kronecker (unsigned long @var{a}, const mpz_t @var{b})
@cindex Kronecker symbol functions
Calculate the Jacobi symbol @m{\left(a \over b\right),
(@var{a}/@var{b})} with the Kronecker extension @m{\left(a \over
2\right) = \left(2 \over a\right), (a/2)=(2/a)} when @math{a} odd, or
@m{\left(a \over 2\right) = 0, (a/2)=0} when @math{a} even.

When @var{b} is odd the Jacobi symbol and Kronecker symbol are
identical, so @code{mpz_kronecker_ui} etc can be used for mixed
precision Jacobi symbols too.

For more information see Henri Cohen section 1.4.2 (@pxref{References}),
or any number theory textbook.  See also the example program
@file{demos/qcn.c} which uses @code{mpz_kronecker_ui}.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpz_remove (mpz_t @var{rop}, const mpz_t @var{op}, const mpz_t @var{f})
@cindex Remove factor functions
@cindex Factor removal functions
Remove all occurrences of the factor @var{f} from @var{op} and store the
result in @var{rop}.  The return value is how many such occurrences were
removed.
@end deftypefun

@deftypefun void mpz_fac_ui (mpz_t @var{rop}, unsigned long int @var{n})
@deftypefunx void mpz_2fac_ui (mpz_t @var{rop}, unsigned long int @var{n})
@deftypefunx void mpz_mfac_uiui (mpz_t @var{rop}, unsigned long int @var{n}, unsigned long int @var{m})
@cindex Factorial functions
Set @var{rop} to the factorial of @var{n}: @code{mpz_fac_ui} computes the plain factorial @var{n}!,
@code{mpz_2fac_ui} computes the double-factorial @var{n}!!, and @code{mpz_mfac_uiui} the
@var{m}-multi-factorial @m{n!^{(m)}, @var{n}!^(@var{m})}.
@end deftypefun

@deftypefun void mpz_primorial_ui (mpz_t @var{rop}, unsigned long int @var{n})
@cindex Primorial functions
Set @var{rop} to the primorial of @var{n}, i.e. the product of all positive
prime numbers @math{@le{}@var{n}}.
@end deftypefun

@deftypefun void mpz_bin_ui (mpz_t @var{rop}, const mpz_t @var{n}, unsigned long int @var{k})
@deftypefunx void mpz_bin_uiui (mpz_t @var{rop}, unsigned long int @var{n}, @w{unsigned long int @var{k}})
@cindex Binomial coefficient functions
Compute the binomial coefficient @m{\left({n}\atop{k}\right), @var{n} over
@var{k}} and store the result in @var{rop}.  Negative values of @var{n} are
supported by @code{mpz_bin_ui}, using the identity
@m{\left({-n}\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right),
bin(-n@C{}k) = (-1)^k * bin(n+k-1@C{}k)}, see Knuth volume 1 section 1.2.6
part G.
@end deftypefun

@deftypefun void mpz_fib_ui (mpz_t @var{fn}, unsigned long int @var{n})
@deftypefunx void mpz_fib2_ui (mpz_t @var{fn}, mpz_t @var{fnsub1}, unsigned long int @var{n})
@cindex Fibonacci sequence functions
@code{mpz_fib_ui} sets @var{fn} to to @m{F_n,F[n]}, the @var{n}'th Fibonacci
number.  @code{mpz_fib2_ui} sets @var{fn} to @m{F_n,F[n]}, and @var{fnsub1} to
@m{F_{n-1},F[n-1]}.

These functions are designed for calculating isolated Fibonacci numbers.  When
a sequence of values is wanted it's best to start with @code{mpz_fib2_ui} and
iterate the defining @m{F_{n+1} = F_n + F_{n-1}, F[n+1]=F[n]+F[n-1]} or
similar.
@end deftypefun

@deftypefun void mpz_lucnum_ui (mpz_t @var{ln}, unsigned long int @var{n})
@deftypefunx void mpz_lucnum2_ui (mpz_t @var{ln}, mpz_t @var{lnsub1}, unsigned long int @var{n})
@cindex Lucas number functions
@code{mpz_lucnum_ui} sets @var{ln} to to @m{L_n,L[n]}, the @var{n}'th Lucas
number.  @code{mpz_lucnum2_ui} sets @var{ln} to @m{L_n,L[n]}, and @var{lnsub1}
to @m{L_{n-1},L[n-1]}.

These functions are designed for calculating isolated Lucas numbers.  When a
sequence of values is wanted it's best to start with @code{mpz_lucnum2_ui} and
iterate the defining @m{L_{n+1} = L_n + L_{n-1}, L[n+1]=L[n]+L[n-1]} or
similar.

The Fibonacci numbers and Lucas numbers are related sequences, so it's never
necessary to call both @code{mpz_fib2_ui} and @code{mpz_lucnum2_ui}.  The
formulas for going from Fibonacci to Lucas can be found in @ref{Lucas Numbers
Algorithm}, the reverse is straightforward too.
@end deftypefun


@node Integer Comparisons, Integer Logic and Bit Fiddling, Number Theoretic Functions, Integer Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Integer comparison functions
@cindex Comparison functions

@deftypefn Function int mpz_cmp (const mpz_t @var{op1}, const mpz_t @var{op2})
@deftypefnx Function int mpz_cmp_d (const mpz_t @var{op1}, double @var{op2})
@deftypefnx Macro int mpz_cmp_si (const mpz_t @var{op1}, signed long int @var{op2})
@deftypefnx Macro int mpz_cmp_ui (const mpz_t @var{op1}, unsigned long int @var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, or a negative value if
@math{@var{op1} < @var{op2}}.

@code{mpz_cmp_ui} and @code{mpz_cmp_si} are macros and will evaluate their
arguments more than once.  @code{mpz_cmp_d} can be called with an infinity,
but results are undefined for a NaN.
@end deftypefn

@deftypefn Function int mpz_cmpabs (const mpz_t @var{op1}, const mpz_t @var{op2})
@deftypefnx Function int mpz_cmpabs_d (const mpz_t @var{op1}, double @var{op2})
@deftypefnx Function int mpz_cmpabs_ui (const mpz_t @var{op1}, unsigned long int @var{op2})
Compare the absolute values of @var{op1} and @var{op2}.  Return a positive
value if @math{@GMPabs{@var{op1}} > @GMPabs{@var{op2}}}, zero if
@math{@GMPabs{@var{op1}} = @GMPabs{@var{op2}}}, or a negative value if
@math{@GMPabs{@var{op1}} < @GMPabs{@var{op2}}}.

@code{mpz_cmpabs_d} can be called with an infinity, but results are undefined
for a NaN.
@end deftypefn

@deftypefn Macro int mpz_sgn (const mpz_t @var{op})
@cindex Sign tests
@cindex Integer sign tests
Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@var{op} < 0}.

This function is actually implemented as a macro.  It evaluates its argument
multiple times.
@end deftypefn


@node Integer Logic and Bit Fiddling, I/O of Integers, Integer Comparisons, Integer Functions
@comment  node-name,  next,  previous,  up
@section Logical and Bit Manipulation Functions
@cindex Logical functions
@cindex Bit manipulation functions
@cindex Integer logical functions
@cindex Integer bit manipulation functions

These functions behave as if twos complement arithmetic were used (although
sign-magnitude is the actual implementation).  The least significant bit is
number 0.

@deftypefun void mpz_and (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
Set @var{rop} to @var{op1} bitwise-and @var{op2}.
@end deftypefun

@deftypefun void mpz_ior (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
Set @var{rop} to @var{op1} bitwise inclusive-or @var{op2}.
@end deftypefun

@deftypefun void mpz_xor (mpz_t @var{rop}, const mpz_t @var{op1}, const mpz_t @var{op2})
Set @var{rop} to @var{op1} bitwise exclusive-or @var{op2}.
@end deftypefun

@deftypefun void mpz_com (mpz_t @var{rop}, const mpz_t @var{op})
Set @var{rop} to the one's complement of @var{op}.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpz_popcount (const mpz_t @var{op})
If @math{@var{op}@ge{}0}, return the population count of @var{op}, which is the
number of 1 bits in the binary representation.  If @math{@var{op}<0}, the
number of 1s is infinite, and the return value is the largest possible
@code{mp_bitcnt_t}.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpz_hamdist (const mpz_t @var{op1}, const mpz_t @var{op2})
If @var{op1} and @var{op2} are both @math{@ge{}0} or both @math{<0}, return the
hamming distance between the two operands, which is the number of bit positions
where @var{op1} and @var{op2} have different bit values.  If one operand is
@math{@ge{}0} and the other @math{<0} then the number of bits different is
infinite, and the return value is the largest possible @code{mp_bitcnt_t}.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpz_scan0 (const mpz_t @var{op}, mp_bitcnt_t @var{starting_bit})
@deftypefunx {mp_bitcnt_t} mpz_scan1 (const mpz_t @var{op}, mp_bitcnt_t @var{starting_bit})
@cindex Bit scanning functions
@cindex Scan bit functions
Scan @var{op}, starting from bit @var{starting_bit}, towards more significant
bits, until the first 0 or 1 bit (respectively) is found.  Return the index of
the found bit.

If the bit at @var{starting_bit} is already what's sought, then
@var{starting_bit} is returned.

If there's no bit found, then the largest possible @code{mp_bitcnt_t} is
returned.  This will happen in @code{mpz_scan0} past the end of a negative
number, or @code{mpz_scan1} past the end of a nonnegative number.
@end deftypefun

@deftypefun void mpz_setbit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
Set bit @var{bit_index} in @var{rop}.
@end deftypefun

@deftypefun void mpz_clrbit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
Clear bit @var{bit_index} in @var{rop}.
@end deftypefun

@deftypefun void mpz_combit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index})
Complement bit @var{bit_index} in @var{rop}.
@end deftypefun

@deftypefun int mpz_tstbit (const mpz_t @var{op}, mp_bitcnt_t @var{bit_index})
Test bit @var{bit_index} in @var{op} and return 0 or 1 accordingly.
@end deftypefun

@node I/O of Integers, Integer Random Numbers, Integer Logic and Bit Fiddling, Integer Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Integer input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

Functions that perform input from a stdio stream, and functions that output to
a stdio stream, of @code{mpz} numbers.  Passing a @code{NULL} pointer for a
@var{stream} argument to any of these functions will make them read from
@code{stdin} and write to @code{stdout}, respectively.

When using any of these functions, it is a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.

See also @ref{Formatted Output} and @ref{Formatted Input}.

@deftypefun size_t mpz_out_str (FILE *@var{stream}, int @var{base}, const mpz_t @var{op})
Output @var{op} on stdio stream @var{stream}, as a string of digits in base
@var{base}.  The base argument may vary from 2 to 62 or from @minus{}2 to
@minus{}36.

For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.

Return the number of bytes written, or if an error occurred, return 0.
@end deftypefun

@deftypefun size_t mpz_inp_str (mpz_t @var{rop}, FILE *@var{stream}, int @var{base})
Input a possibly white-space preceded string in base @var{base} from stdio
stream @var{stream}, and put the read integer in @var{rop}.

The @var{base} may vary from 2 to 62, or if @var{base} is 0, then the leading
characters are used: @code{0x} and @code{0X} for hexadecimal, @code{0b} and
@code{0B} for binary, @code{0} for octal, or decimal otherwise.

For bases up to 36, case is ignored; upper-case and lower-case letters have
the same value.  For bases 37 to 62, upper-case letter represent the usual
10..35 while lower-case letter represent 36..61.

Return the number of bytes read, or if an error occurred, return 0.
@end deftypefun

@deftypefun size_t mpz_out_raw (FILE *@var{stream}, const mpz_t @var{op})
Output @var{op} on stdio stream @var{stream}, in raw binary format.  The
integer is written in a portable format, with 4 bytes of size information, and
that many bytes of limbs.  Both the size and the limbs are written in
decreasing significance order (i.e., in big-endian).

The output can be read with @code{mpz_inp_raw}.

Return the number of bytes written, or if an error occurred, return 0.

The output of this can not be read by @code{mpz_inp_raw} from GMP 1, because
of changes necessary for compatibility between 32-bit and 64-bit machines.
@end deftypefun

@deftypefun size_t mpz_inp_raw (mpz_t @var{rop}, FILE *@var{stream})
Input from stdio stream @var{stream} in the format written by
@code{mpz_out_raw}, and put the result in @var{rop}.  Return the number of
bytes read, or if an error occurred, return 0.

This routine can read the output from @code{mpz_out_raw} also from GMP 1, in
spite of changes necessary for compatibility between 32-bit and 64-bit
machines.
@end deftypefun


@need 2000
@node Integer Random Numbers, Integer Import and Export, I/O of Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Random Number Functions
@cindex Integer random number functions
@cindex Random number functions

The random number functions of GMP come in two groups; older function
that rely on a global state, and newer functions that accept a state
parameter that is read and modified.  Please see the @ref{Random Number
Functions} for more information on how to use and not to use random
number functions.

@deftypefun void mpz_urandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{n})
Generate a uniformly distributed random integer in the range 0 to
@mm{2@sup{n}-1, 2^@var{n}@minus{}1}, inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization}) before
invoking this function.
@end deftypefun

@deftypefun void mpz_urandomm (mpz_t @var{rop}, gmp_randstate_t @var{state}, const mpz_t @var{n})
Generate a uniform random integer in the range 0 to @math{@var{n}-1},
inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization})
before invoking this function.
@end deftypefun

@deftypefun void mpz_rrandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{n})
Generate a random integer with long strings of zeros and ones in the
binary representation.  Useful for testing functions and algorithms,
since this kind of random numbers have proven to be more likely to
trigger corner-case bugs.  The random number will be in the range
@mm{2@sup{n-1}, 2^(@var{n}@minus{}1)} to @mm{2@sup{n}-1,
2^@var{n}@minus{}1}, inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization})
before invoking this function.
@end deftypefun

@deftypefun void mpz_random (mpz_t @var{rop}, mp_size_t @var{max_size})
Generate a random integer of at most @var{max_size} limbs.  The generated
random number doesn't satisfy any particular requirements of randomness.
Negative random numbers are generated when @var{max_size} is negative.

This function is obsolete.  Use @code{mpz_urandomb} or
@code{mpz_urandomm} instead.
@end deftypefun

@deftypefun void mpz_random2 (mpz_t @var{rop}, mp_size_t @var{max_size})
Generate a random integer of at most @var{max_size} limbs, with long strings
of zeros and ones in the binary representation.  Useful for testing functions
and algorithms, since this kind of random numbers have proven to be more
likely to trigger corner-case bugs.  Negative random numbers are generated
when @var{max_size} is negative.

This function is obsolete.  Use @code{mpz_rrandomb} instead.
@end deftypefun


@node Integer Import and Export, Miscellaneous Integer Functions, Integer Random Numbers, Integer Functions
@section Integer Import and Export

@code{mpz_t} variables can be converted to and from arbitrary words of binary
data with the following functions.

@deftypefun void mpz_import (mpz_t @var{rop}, size_t @var{count}, int @var{order}, size_t @var{size}, int @var{endian}, size_t @var{nails}, const void *@var{op})
@cindex Integer import
@cindex Import
Set @var{rop} from an array of word data at @var{op}.

The parameters specify the format of the data.  @var{count} many words are
read, each @var{size} bytes.  @var{order} can be 1 for most significant word
first or -1 for least significant first.  Within each word @var{endian} can be
1 for most significant byte first, -1 for least significant first, or 0 for
the native endianness of the host CPU@.  The most significant @var{nails} bits
of each word are skipped, this can be 0 to use the full words.

There is no sign taken from the data, @var{rop} will simply be a positive
integer.  An application can handle any sign itself, and apply it for instance
with @code{mpz_neg}.

There are no data alignment restrictions on @var{op}, any address is allowed.

Here's an example converting an array of @code{unsigned long} data, most
significant element first, and host byte order within each value.

@example
unsigned long  a[20];
/* Initialize @var{z} and @var{a} */
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
@end example

This example assumes the full @code{sizeof} bytes are used for data in the
given type, which is usually true, and certainly true for @code{unsigned long}
everywhere we know of.  However on Cray vector systems it may be noted that
@code{short} and @code{int} are always stored in 8 bytes (and with
@code{sizeof} indicating that) but use only 32 or 46 bits.  The @var{nails}
feature can account for this, by passing for instance
@code{8*sizeof(int)-INT_BIT}.
@end deftypefun

@deftypefun {void *} mpz_export (void *@var{rop}, size_t *@var{countp}, int @var{order}, size_t @var{size}, int @var{endian}, size_t @var{nails}, const mpz_t @var{op})
@cindex Integer export
@cindex Export
Fill @var{rop} with word data from @var{op}.

The parameters specify the format of the data produced.  Each word will be
@var{size} bytes and @var{order} can be 1 for most significant word first or
-1 for least significant first.  Within each word @var{endian} can be 1 for
most significant byte first, -1 for least significant first, or 0 for the
native endianness of the host CPU@.  The most significant @var{nails} bits of
each word are unused and set to zero, this can be 0 to produce full words.

The number of words produced is written to @code{*@var{countp}}, or
@var{countp} can be @code{NULL} to discard the count.  @var{rop} must have
enough space for the data, or if @var{rop} is @code{NULL} then a result array
of the necessary size is allocated using the current GMP allocation function
(@pxref{Custom Allocation}).  In either case the return value is the
destination used, either @var{rop} or the allocated block.

If @var{op} is non-zero then the most significant word produced will be
non-zero.  If @var{op} is zero then the count returned will be zero and
nothing written to @var{rop}.  If @var{rop} is @code{NULL} in this case, no
block is allocated, just @code{NULL} is returned.

The sign of @var{op} is ignored, just the absolute value is exported.  An
application can use @code{mpz_sgn} to get the sign and handle it as desired.
(@pxref{Integer Comparisons})

There are no data alignment restrictions on @var{rop}, any address is allowed.

When an application is allocating space itself the required size can be
determined with a calculation like the following.  Since @code{mpz_sizeinbase}
always returns at least 1, @code{count} here will be at least one, which
avoids any portability problems with @code{malloc(0)}, though if @code{z} is
zero no space at all is actually needed (or written).

@example
numb = 8*size - nail;
count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
p = malloc (count * size);
@end example
@end deftypefun


@need 2000
@node Miscellaneous Integer Functions, Integer Special Functions, Integer Import and Export, Integer Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous integer functions
@cindex Integer miscellaneous functions

@deftypefun int mpz_fits_ulong_p (const mpz_t @var{op})
@deftypefunx int mpz_fits_slong_p (const mpz_t @var{op})
@deftypefunx int mpz_fits_uint_p (const mpz_t @var{op})
@deftypefunx int mpz_fits_sint_p (const mpz_t @var{op})
@deftypefunx int mpz_fits_ushort_p (const mpz_t @var{op})
@deftypefunx int mpz_fits_sshort_p (const mpz_t @var{op})
Return non-zero iff the value of @var{op} fits in an @code{unsigned long int},
@code{signed long int}, @code{unsigned int}, @code{signed int}, @code{unsigned
short int}, or @code{signed short int}, respectively.  Otherwise, return zero.
@end deftypefun

@deftypefn Macro int mpz_odd_p (const mpz_t @var{op})
@deftypefnx Macro int mpz_even_p (const mpz_t @var{op})
Determine whether @var{op} is odd or even, respectively.  Return non-zero if
yes, zero if no.  These macros evaluate their argument more than once.
@end deftypefn

@deftypefun size_t mpz_sizeinbase (const mpz_t @var{op}, int @var{base})
@cindex Size in digits
@cindex Digits in an integer
Return the size of @var{op} measured in number of digits in the given
@var{base}.  @var{base} can vary from 2 to 62.  The sign of @var{op} is
ignored, just the absolute value is used.  The result will be either exact or
1 too big.  If @var{base} is a power of 2, the result is always exact.  If
@var{op} is zero the return value is always 1.

This function can be used to determine the space required when converting
@var{op} to a string.  The right amount of allocation is normally two more
than the value returned by @code{mpz_sizeinbase}, one extra for a minus sign
and one for the null-terminator.

@cindex Most significant bit
It will be noted that @code{mpz_sizeinbase(@var{op},2)} can be used to locate
the most significant 1 bit in @var{op}, counting from 1.  (Unlike the bitwise
functions which start from 0, @xref{Integer Logic and Bit Fiddling,, Logical
and Bit Manipulation Functions}.)
@end deftypefun


@node Integer Special Functions,  , Miscellaneous Integer Functions, Integer Functions
@section Special Functions
@cindex Special integer functions
@cindex Integer special functions

The functions in this section are for various special purposes.  Most
applications will not need them.

@deftypefun void mpz_array_init (mpz_t @var{integer_array}, mp_size_t @var{array_size}, @w{mp_size_t @var{fixed_num_bits}})
@strong{This is an obsolete function.  Do not use it.}
@end deftypefun

@deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc})
Change the space for @var{integer} to @var{new_alloc} limbs.  The value in
@var{integer} is preserved if it fits, or is set to 0 if not.  The return
value is not useful to applications and should be ignored.

@code{mpz_realloc2} is the preferred way to accomplish allocation changes like
this.  @code{mpz_realloc2} and @code{_mpz_realloc} are the same except that
@code{_mpz_realloc} takes its size in limbs.
@end deftypefun

@deftypefun mp_limb_t mpz_getlimbn (const mpz_t @var{op}, mp_size_t @var{n})
Return limb number @var{n} from @var{op}.  The sign of @var{op} is ignored,
just the absolute value is used.  The least significant limb is number 0.

@code{mpz_size} can be used to find how many limbs make up @var{op}.
@code{mpz_getlimbn} returns zero if @var{n} is outside the range 0 to
@code{mpz_size(@var{op})-1}.
@end deftypefun

@deftypefun size_t mpz_size (const mpz_t @var{op})
Return the size of @var{op} measured in number of limbs.  If @var{op} is zero,
the returned value will be zero.
@c (@xref{Nomenclature}, for an explanation of the concept @dfn{limb}.)
@end deftypefun

@deftypefun {const mp_limb_t *} mpz_limbs_read (const mpz_t @var{x})
Return a pointer to the limb array representing the absolute value of @var{x}.
The size of the array is @code{mpz_size(@var{x})}. Intended for read access
only.
@end deftypefun

@deftypefun {mp_limb_t *} mpz_limbs_write (mpz_t @var{x}, mp_size_t @var{n})
@deftypefunx {mp_limb_t *} mpz_limbs_modify (mpz_t @var{x}, mp_size_t @var{n})
Return a pointer to the limb array, intended for write access. The array is
reallocated as needed, to make room for @var{n} limbs. Requires @math{@var{n}
> 0}. The @code{mpz_limbs_modify} function returns an array that holds the old
absolute value of @var{x}, while @code{mpz_limbs_write} may destroy the old
value and return an array with unspecified contents.
@end deftypefun

@deftypefun void mpz_limbs_finish (mpz_t @var{x}, mp_size_t @var{s})
Updates the internal size field of @var{x}. Used after writing to the limb
array pointer returned by @code{mpz_limbs_write} or @code{mpz_limbs_modify} is
completed. The array should contain @math{@GMPabs{@var{s}}} valid limbs,
representing the new absolute value for @var{x}, and the sign of @var{x} is
taken from the sign of @var{s}. This function never reallocates @var{x}, so
the limb pointer remains valid.
@end deftypefun

@c FIXME: Some more useful and less silly example?
@example
void foo (mpz_t x)
@{
  mp_size_t n, i;
  mp_limb_t *xp;

  n = mpz_size (x);
  xp = mpz_limbs_modify (x, 2*n);
  for (i = 0; i < n; i++)
    xp[n+i] = xp[n-1-i];
  mpz_limbs_finish (x, mpz_sgn (x) < 0 ? - 2*n : 2*n);
@}
@end example

@deftypefun mpz_srcptr mpz_roinit_n (mpz_t @var{x}, const mp_limb_t *@var{xp}, mp_size_t @var{xs})
Special initialization of @var{x}, using the given limb array and size.
@var{x} should be treated as read-only: it can be passed safely as input to
any mpz function, but not as an output. The array @var{xp} must point to at
least a readable limb, its size is
@math{@GMPabs{@var{xs}}}, and the sign of @var{x} is the sign of @var{xs}. For
convenience, the function returns @var{x}, but cast to a const pointer type.
@end deftypefun

@example
void foo (mpz_t x)
@{
  static const mp_limb_t y[3] = @{ 0x1, 0x2, 0x3 @};
  mpz_t tmp;
  mpz_add (x, x, mpz_roinit_n (tmp, y, 3));
@}
@end example

@deftypefn Macro mpz_t MPZ_ROINIT_N (mp_limb_t *@var{xp}, mp_size_t @var{xs})
This macro expands to an initializer which can be assigned to an mpz_t
variable. The limb array @var{xp} must point to at least a readable limb,
moreover, unlike the @code{mpz_roinit_n} function, the array must be
normalized: if @var{xs} is non-zero, then
@code{@var{xp}[@math{@GMPabs{@var{xs}}-1}]} must be non-zero. Intended
primarily for constant values. Using it for non-constant values requires a C
compiler supporting C99.
@end deftypefn

@example
void foo (mpz_t x)
@{
  static const mp_limb_t ya[3] = @{ 0x1, 0x2, 0x3 @};
  static const mpz_t y = MPZ_ROINIT_N ((mp_limb_t *) ya, 3);

  mpz_add (x, x, y);
@}
@end example


@node Rational Number Functions, Floating-point Functions, Integer Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Rational Number Functions
@cindex Rational number functions

This chapter describes the GMP functions for performing arithmetic on rational
numbers.  These functions start with the prefix @code{mpq_}.

Rational numbers are stored in objects of type @code{mpq_t}.

All rational arithmetic functions assume operands have a canonical form, and
canonicalize their result.  The canonical form means that the denominator and
the numerator have no common factors, and that the denominator is positive.
Zero has the unique representation 0/1.

Pure assignment functions do not canonicalize the assigned variable.  It is
the responsibility of the user to canonicalize the assigned variable before
any arithmetic operations are performed on that variable.

@deftypefun void mpq_canonicalize (mpq_t @var{op})
Remove any factors that are common to the numerator and denominator of
@var{op}, and make the denominator positive.
@end deftypefun

@menu
* Initializing Rationals::
* Rational Conversions::
* Rational Arithmetic::
* Comparing Rationals::
* Applying Integer Functions::
* I/O of Rationals::
@end menu

@node Initializing Rationals, Rational Conversions, Rational Number Functions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Initialization and Assignment Functions
@cindex Rational assignment functions
@cindex Assignment functions
@cindex Rational initialization functions
@cindex Initialization functions

@deftypefun void mpq_init (mpq_t @var{x})
Initialize @var{x} and set it to 0/1.  Each variable should normally only be
initialized once, or at least cleared out (using the function @code{mpq_clear})
between each initialization.
@end deftypefun

@deftypefun void mpq_inits (mpq_t @var{x}, ...)
Initialize a NULL-terminated list of @code{mpq_t} variables, and set their
values to 0/1.
@end deftypefun

@deftypefun void mpq_clear (mpq_t @var{x})
Free the space occupied by @var{x}.  Make sure to call this function for all
@code{mpq_t} variables when you are done with them.
@end deftypefun

@deftypefun void mpq_clears (mpq_t @var{x}, ...)
Free the space occupied by a NULL-terminated list of @code{mpq_t} variables.
@end deftypefun

@deftypefun void mpq_set (mpq_t @var{rop}, const mpq_t @var{op})
@deftypefunx void mpq_set_z (mpq_t @var{rop}, const mpz_t @var{op})
Assign @var{rop} from @var{op}.
@end deftypefun

@deftypefun void mpq_set_ui (mpq_t @var{rop}, unsigned long int @var{op1}, unsigned long int @var{op2})
@deftypefunx void mpq_set_si (mpq_t @var{rop}, signed long int @var{op1}, unsigned long int @var{op2})
Set the value of @var{rop} to @var{op1}/@var{op2}.  Note that if @var{op1} and
@var{op2} have common factors, @var{rop} has to be passed to
@code{mpq_canonicalize} before any operations are performed on @var{rop}.
@end deftypefun

@deftypefun int mpq_set_str (mpq_t @var{rop}, const char *@var{str}, int @var{base})
Set @var{rop} from a null-terminated string @var{str} in the given @var{base}.

The string can be an integer like ``41'' or a fraction like ``41/152''.  The
fraction must be in canonical form (@pxref{Rational Number Functions}), or if
not then @code{mpq_canonicalize} must be called.

The numerator and optional denominator are parsed the same as in
@code{mpz_set_str} (@pxref{Assigning Integers}).  White space is allowed in
the string, and is simply ignored.  The @var{base} can vary from 2 to 62, or
if @var{base} is 0 then the leading characters are used: @code{0x} or @code{0X} for hex,
@code{0b} or @code{0B} for binary,
@code{0} for octal, or decimal otherwise.  Note that this is done separately
for the numerator and denominator, so for instance @code{0xEF/100} is 239/100,
whereas @code{0xEF/0x100} is 239/256.

The return value is 0 if the entire string is a valid number, or @minus{}1 if
not.
@end deftypefun

@deftypefun void mpq_swap (mpq_t @var{rop1}, mpq_t @var{rop2})
Swap the values @var{rop1} and @var{rop2} efficiently.
@end deftypefun


@need 2000
@node Rational Conversions, Rational Arithmetic, Initializing Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Rational conversion functions
@cindex Conversion functions

@deftypefun double mpq_get_d (const mpq_t @var{op})
Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding
towards zero).

If the exponent from the conversion is too big or too small to fit a
@code{double} then the result is system dependent.  For too big an infinity is
returned when available.  For too small @math{0.0} is normally returned.
Hardware overflow, underflow and denorm traps may or may not occur.
@end deftypefun

@deftypefun void mpq_set_d (mpq_t @var{rop}, double @var{op})
@deftypefunx void mpq_set_f (mpq_t @var{rop}, const mpf_t @var{op})
Set @var{rop} to the value of @var{op}.  There is no rounding, this conversion
is exact.
@end deftypefun

@deftypefun {char *} mpq_get_str (char *@var{str}, int @var{base}, const mpq_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  The base argument
may vary from 2 to 62 or from @minus{}2 to @minus{}36.  The string will be of
the form @samp{num/den}, or if the denominator is 1 then just @samp{num}.

For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of storage large
enough for the result, that being

@example
mpz_sizeinbase (mpq_numref(@var{op}), @var{base})
+ mpz_sizeinbase (mpq_denref(@var{op}), @var{base}) + 3
@end example

The three extra bytes are for a possible minus sign, possible slash, and the
null-terminator.

A pointer to the result string is returned, being either the allocated block,
or the given @var{str}.
@end deftypefun


@node Rational Arithmetic, Comparing Rationals, Rational Conversions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Rational arithmetic functions
@cindex Arithmetic functions

@deftypefun void mpq_add (mpq_t @var{sum}, const mpq_t @var{addend1}, const mpq_t @var{addend2})
Set @var{sum} to @var{addend1} + @var{addend2}.
@end deftypefun

@deftypefun void mpq_sub (mpq_t @var{difference}, const mpq_t @var{minuend}, const mpq_t @var{subtrahend})
Set @var{difference} to @var{minuend} @minus{} @var{subtrahend}.
@end deftypefun

@deftypefun void mpq_mul (mpq_t @var{product}, const mpq_t @var{multiplier}, const mpq_t @var{multiplicand})
Set @var{product} to @math{@var{multiplier} @GMPtimes{} @var{multiplicand}}.
@end deftypefun

@deftypefun void mpq_mul_2exp (mpq_t @var{rop}, const mpq_t @var{op1}, mp_bitcnt_t @var{op2})
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.
@end deftypefun

@deftypefun void mpq_div (mpq_t @var{quotient}, const mpq_t @var{dividend}, const mpq_t @var{divisor})
@cindex Division functions
Set @var{quotient} to @var{dividend}/@var{divisor}.
@end deftypefun

@deftypefun void mpq_div_2exp (mpq_t @var{rop}, const mpq_t @var{op1}, mp_bitcnt_t @var{op2})
Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
@var{op2}}.
@end deftypefun

@deftypefun void mpq_neg (mpq_t @var{negated_operand}, const mpq_t @var{operand})
Set @var{negated_operand} to @minus{}@var{operand}.
@end deftypefun

@deftypefun void mpq_abs (mpq_t @var{rop}, const mpq_t @var{op})
Set @var{rop} to the absolute value of @var{op}.
@end deftypefun

@deftypefun void mpq_inv (mpq_t @var{inverted_number}, const mpq_t @var{number})
Set @var{inverted_number} to 1/@var{number}.  If the new denominator is
zero, this routine will divide by zero.
@end deftypefun

@node Comparing Rationals, Applying Integer Functions, Rational Arithmetic, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Rational comparison functions
@cindex Comparison functions

@deftypefun int mpq_cmp (const mpq_t @var{op1}, const mpq_t @var{op2})
@deftypefunx int mpq_cmp_z (const mpq_t @var{op1}, const mpz_t @var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
@math{@var{op1} < @var{op2}}.

To determine if two rationals are equal, @code{mpq_equal} is faster than
@code{mpq_cmp}.
@end deftypefun

@deftypefn Macro int mpq_cmp_ui (const mpq_t @var{op1}, unsigned long int @var{num2}, unsigned long int @var{den2})
@deftypefnx Macro int mpq_cmp_si (const mpq_t @var{op1}, long int @var{num2}, unsigned long int @var{den2})
Compare @var{op1} and @var{num2}/@var{den2}.  Return a positive value if
@math{@var{op1} > @var{num2}/@var{den2}}, zero if @math{@var{op1} =
@var{num2}/@var{den2}}, and a negative value if @math{@var{op1} <
@var{num2}/@var{den2}}.

@var{num2} and @var{den2} are allowed to have common factors.

These functions are implemented as a macros and evaluate their arguments
multiple times.
@end deftypefn

@deftypefn Macro int mpq_sgn (const mpq_t @var{op})
@cindex Sign tests
@cindex Rational sign tests
Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@var{op} < 0}.

This function is actually implemented as a macro.  It evaluates its
argument multiple times.
@end deftypefn

@deftypefun int mpq_equal (const mpq_t @var{op1}, const mpq_t @var{op2})
Return non-zero if @var{op1} and @var{op2} are equal, zero if they are
non-equal.  Although @code{mpq_cmp} can be used for the same purpose, this
function is much faster.
@end deftypefun

@node Applying Integer Functions, I/O of Rationals, Comparing Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Applying Integer Functions to Rationals
@cindex Rational numerator and denominator
@cindex Numerator and denominator

The set of @code{mpq} functions is quite small.  In particular, there are few
functions for either input or output.  The following functions give direct
access to the numerator and denominator of an @code{mpq_t}.

Note that if an assignment to the numerator and/or denominator could take an
@code{mpq_t} out of the canonical form described at the start of this chapter
(@pxref{Rational Number Functions}) then @code{mpq_canonicalize} must be
called before any other @code{mpq} functions are applied to that @code{mpq_t}.

@deftypefn Macro mpz_t mpq_numref (const mpq_t @var{op})
@deftypefnx Macro mpz_t mpq_denref (const mpq_t @var{op})
Return a reference to the numerator and denominator of @var{op}, respectively.
The @code{mpz} functions can be used on the result of these macros.
@end deftypefn

@deftypefun void mpq_get_num (mpz_t @var{numerator}, const mpq_t @var{rational})
@deftypefunx void mpq_get_den (mpz_t @var{denominator}, const mpq_t @var{rational})
@deftypefunx void mpq_set_num (mpq_t @var{rational}, const mpz_t @var{numerator})
@deftypefunx void mpq_set_den (mpq_t @var{rational}, const mpz_t @var{denominator})
Get or set the numerator or denominator of a rational.  These functions are
equivalent to calling @code{mpz_set} with an appropriate @code{mpq_numref} or
@code{mpq_denref}.  Direct use of @code{mpq_numref} or @code{mpq_denref} is
recommended instead of these functions.
@end deftypefun


@need 2000
@node I/O of Rationals,  , Applying Integer Functions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Rational input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

Functions that perform input from a stdio stream, and functions that output to
a stdio stream, of @code{mpq} numbers.  Passing a @code{NULL} pointer for a
@var{stream} argument to any of these functions will make them read from
@code{stdin} and write to @code{stdout}, respectively.

When using any of these functions, it is a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.

See also @ref{Formatted Output} and @ref{Formatted Input}.

@deftypefun size_t mpq_out_str (FILE *@var{stream}, int @var{base}, const mpq_t @var{op})
Output @var{op} on stdio stream @var{stream}, as a string of digits in base
@var{base}.  The base argument may vary from 2 to 62 or from @minus{}2 to
@minus{}36. Output is in the form
@samp{num/den} or if the denominator is 1 then just @samp{num}.

For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.

Return the number of bytes written, or if an error occurred, return 0.
@end deftypefun

@deftypefun size_t mpq_inp_str (mpq_t @var{rop}, FILE *@var{stream}, int @var{base})
Read a string of digits from @var{stream} and convert them to a rational in
@var{rop}.  Any initial white-space characters are read and discarded.  Return
the number of characters read (including white space), or 0 if a rational
could not be read.

The input can be a fraction like @samp{17/63} or just an integer like
@samp{123}.  Reading stops at the first character not in this form, and white
space is not permitted within the string.  If the input might not be in
canonical form, then @code{mpq_canonicalize} must be called (@pxref{Rational
Number Functions}).

The @var{base} can be between 2 and 62, or can be 0 in which case the leading
characters of the string determine the base, @samp{0x} or @samp{0X} for
hexadecimal, @code{0b} and @code{0B} for binary, @samp{0} for octal, or
decimal otherwise.  The leading characters
are examined separately for the numerator and denominator of a fraction, so
for instance @samp{0x10/11} is @math{16/11}, whereas @samp{0x10/0x11} is
@math{16/17}.
@end deftypefun


@node Floating-point Functions, Low-level Functions, Rational Number Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Floating-point Functions
@cindex Floating-point functions
@cindex Float functions
@cindex User-defined precision
@cindex Precision of floats

GMP floating point numbers are stored in objects of type @code{mpf_t} and
functions operating on them have an @code{mpf_} prefix.

The mantissa of each float has a user-selectable precision, in practice only
limited by available memory.  Each variable has its own precision, and that can
be increased or decreased at any time.  This selectable precision is a minimum
value, GMP rounds it up to a whole limb.

The accuracy of a calculation is determined by the priorly set precision of the
destination variable and the numeric values of the input variables.  Input
variables' set precisions do not affect calculations (except indirectly as
their values might have been affected when they were assigned).

The exponent of each float has fixed precision, one machine word on most
systems.  In the current implementation the exponent is a count of limbs, so
for example on a 32-bit system this means a range of roughly
@math{2^@W{-68719476768}} to @math{2^@W{68719476736}}, or on a 64-bit system
this will be much greater.  Note however that @code{mpf_get_str} can only
return an exponent which fits an @code{mp_exp_t} and currently
@code{mpf_set_str} doesn't accept exponents bigger than a @code{long}.

Each variable keeps track of the mantissa data actually in use.  This means
that if a float is exactly represented in only a few bits then only those bits
will be used in a calculation, even if the variable's selected precision is
high.  This is a performance optimization; it does not affect the numeric
results.

Internally, GMP sometimes calculates with higher precision than that of the
destination variable in order to limit errors.  Final results are always
truncated to the destination variable's precision.

The mantissa is stored in binary.  One consequence of this is that decimal
fractions like @math{0.1} cannot be represented exactly.  The same is true of
plain IEEE @code{double} floats.  This makes both highly unsuitable for
calculations involving money or other values that should be exact decimal
fractions.  (Suitably scaled integers, or perhaps rationals, are better
choices.)

The @code{mpf} functions and variables have no special notion of infinity or
not-a-number, and applications must take care not to overflow the exponent or
results will be unpredictable.

Note that the @code{mpf} functions are @emph{not} intended as a smooth
extension to IEEE P754 arithmetic.  In particular results obtained on one
computer often differ from the results on a computer with a different word
size.

New projects should consider using the GMP extension library MPFR
(@url{http://mpfr.org}) instead.  MPFR provides well-defined precision and
accurate rounding, and thereby naturally extends IEEE P754.

@menu
* Initializing Floats::
* Assigning Floats::
* Simultaneous Float Init & Assign::
* Converting Floats::
* Float Arithmetic::
* Float Comparison::
* I/O of Floats::
* Miscellaneous Float Functions::
@end menu

@node Initializing Floats, Assigning Floats, Floating-point Functions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Initialization Functions
@cindex Float initialization functions
@cindex Initialization functions

@deftypefun void mpf_set_default_prec (mp_bitcnt_t @var{prec})
Set the default precision to be @strong{at least} @var{prec} bits.  All
subsequent calls to @code{mpf_init} will use this precision, but previously
initialized variables are unaffected.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpf_get_default_prec (void)
Return the default precision actually used.
@end deftypefun

An @code{mpf_t} object must be initialized before storing the first value in
it.  The functions @code{mpf_init} and @code{mpf_init2} are used for that
purpose.

@deftypefun void mpf_init (mpf_t @var{x})
Initialize @var{x} to 0.  Normally, a variable should be initialized once only
or at least be cleared, using @code{mpf_clear}, between initializations.  The
precision of @var{x} is undefined unless a default precision has already been
established by a call to @code{mpf_set_default_prec}.
@end deftypefun

@deftypefun void mpf_init2 (mpf_t @var{x}, mp_bitcnt_t @var{prec})
Initialize @var{x} to 0 and set its precision to be @strong{at least}
@var{prec} bits.  Normally, a variable should be initialized once only or at
least be cleared, using @code{mpf_clear}, between initializations.
@end deftypefun

@deftypefun void mpf_inits (mpf_t @var{x}, ...)
Initialize a NULL-terminated list of @code{mpf_t} variables, and set their
values to 0.  The precision of the initialized variables is undefined unless a
default precision has already been established by a call to
@code{mpf_set_default_prec}.
@end deftypefun

@deftypefun void mpf_clear (mpf_t @var{x})
Free the space occupied by @var{x}.  Make sure to call this function for all
@code{mpf_t} variables when you are done with them.
@end deftypefun

@deftypefun void mpf_clears (mpf_t @var{x}, ...)
Free the space occupied by a NULL-terminated list of @code{mpf_t} variables.
@end deftypefun

@need 2000
Here is an example on how to initialize floating-point variables:
@example
@{
  mpf_t x, y;
  mpf_init (x);           /* use default precision */
  mpf_init2 (y, 256);     /* precision @emph{at least} 256 bits */
  @dots{}
  /* Unless the program is about to exit, do ... */
  mpf_clear (x);
  mpf_clear (y);
@}
@end example

The following three functions are useful for changing the precision during a
calculation.  A typical use would be for adjusting the precision gradually in
iterative algorithms like Newton-Raphson, making the computation precision
closely match the actual accurate part of the numbers.

@deftypefun {mp_bitcnt_t} mpf_get_prec (const mpf_t @var{op})
Return the current precision of @var{op}, in bits.
@end deftypefun

@deftypefun void mpf_set_prec (mpf_t @var{rop}, mp_bitcnt_t @var{prec})
Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.  The
value in @var{rop} will be truncated to the new precision.

This function requires a call to @code{realloc}, and so should not be used in
a tight loop.
@end deftypefun

@deftypefun void mpf_set_prec_raw (mpf_t @var{rop}, mp_bitcnt_t @var{prec})
Set the precision of @var{rop} to be @strong{at least} @var{prec} bits,
without changing the memory allocated.

@var{prec} must be no more than the allocated precision for @var{rop}, that
being the precision when @var{rop} was initialized, or in the most recent
@code{mpf_set_prec}.

The value in @var{rop} is unchanged, and in particular if it had a higher
precision than @var{prec} it will retain that higher precision.  New values
written to @var{rop} will use the new @var{prec}.

Before calling @code{mpf_clear} or the full @code{mpf_set_prec}, another
@code{mpf_set_prec_raw} call must be made to restore @var{rop} to its original
allocated precision.  Failing to do so will have unpredictable results.

@code{mpf_get_prec} can be used before @code{mpf_set_prec_raw} to get the
original allocated precision.  After @code{mpf_set_prec_raw} it reflects the
@var{prec} value set.

@code{mpf_set_prec_raw} is an efficient way to use an @code{mpf_t} variable at
different precisions during a calculation, perhaps to gradually increase
precision in an iteration, or just to use various different precisions for
different purposes during a calculation.
@end deftypefun


@need 2000
@node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Assignment Functions
@cindex Float assignment functions
@cindex Assignment functions

These functions assign new values to already initialized floats
(@pxref{Initializing Floats}).

@deftypefun void mpf_set (mpf_t @var{rop}, const mpf_t @var{op})
@deftypefunx void mpf_set_ui (mpf_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpf_set_si (mpf_t @var{rop}, signed long int @var{op})
@deftypefunx void mpf_set_d (mpf_t @var{rop}, double @var{op})
@deftypefunx void mpf_set_z (mpf_t @var{rop}, const mpz_t @var{op})
@deftypefunx void mpf_set_q (mpf_t @var{rop}, const mpq_t @var{op})
Set the value of @var{rop} from @var{op}.
@end deftypefun

@deftypefun int mpf_set_str (mpf_t @var{rop}, const char *@var{str}, int @var{base})
Set the value of @var{rop} from the string in @var{str}.  The string is of the
form @samp{M@@N} or, if the base is 10 or less, alternatively @samp{MeN}.
@samp{M} is the mantissa and @samp{N} is the exponent.  The mantissa is always
in the specified base.  The exponent is either in the specified base or, if
@var{base} is negative, in decimal.  The decimal point expected is taken from
the current locale, on systems providing @code{localeconv}.

The argument @var{base} may be in the ranges 2 to 62, or @minus{}62 to
@minus{}2.  Negative values are used to specify that the exponent is in
decimal.

For bases up to 36, case is ignored; upper-case and lower-case letters have
the same value; for bases 37 to 62, upper-case letter represent the usual
10..35 while lower-case letter represent 36..61.

Unlike the corresponding @code{mpz} function, the base will not be determined
from the leading characters of the string if @var{base} is 0.  This is so that
numbers like @samp{0.23} are not interpreted as octal.

White space is allowed in the string, and is simply ignored.  [This is not
really true; white-space is ignored in the beginning of the string and within
the mantissa, but not in other places, such as after a minus sign or in the
exponent.  We are considering changing the definition of this function, making
it fail when there is any white-space in the input, since that makes a lot of
sense.  Please tell us your opinion about this change.  Do you really want it
to accept @nicode{"3 14"} as meaning 314 as it does now?]

This function returns 0 if the entire string is a valid number in base
@var{base}.  Otherwise it returns @minus{}1.
@end deftypefun

@deftypefun void mpf_swap (mpf_t @var{rop1}, mpf_t @var{rop2})
Swap @var{rop1} and @var{rop2} efficiently.  Both the values and the
precisions of the two variables are swapped.
@end deftypefun


@node Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Combined Initialization and Assignment Functions
@cindex Float assignment functions
@cindex Assignment functions
@cindex Float initialization functions
@cindex Initialization functions

For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there.  These functions'
names have the form @code{mpf_init_set@dots{}}

Once the float has been initialized by any of the @code{mpf_init_set@dots{}}
functions, it can be used as the source or destination operand for the ordinary
float functions.  Don't use an initialize-and-set function on a variable
already initialized!

@deftypefun void mpf_init_set (mpf_t @var{rop}, const mpf_t @var{op})
@deftypefunx void mpf_init_set_ui (mpf_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpf_init_set_si (mpf_t @var{rop}, signed long int @var{op})
@deftypefunx void mpf_init_set_d (mpf_t @var{rop}, double @var{op})
Initialize @var{rop} and set its value from @var{op}.

The precision of @var{rop} will be taken from the active default precision, as
set by @code{mpf_set_default_prec}.
@end deftypefun

@deftypefun int mpf_init_set_str (mpf_t @var{rop}, const char *@var{str}, int @var{base})
Initialize @var{rop} and set its value from the string in @var{str}.  See
@code{mpf_set_str} above for details on the assignment operation.

Note that @var{rop} is initialized even if an error occurs.  (I.e., you have to
call @code{mpf_clear} for it.)

The precision of @var{rop} will be taken from the active default precision, as
set by @code{mpf_set_default_prec}.
@end deftypefun


@node Converting Floats, Float Arithmetic, Simultaneous Float Init & Assign, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Float conversion functions
@cindex Conversion functions

@deftypefun double mpf_get_d (const mpf_t @var{op})
Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding
towards zero).

If the exponent in @var{op} is too big or too small to fit a @code{double}
then the result is system dependent.  For too big an infinity is returned when
available.  For too small @math{0.0} is normally returned.  Hardware overflow,
underflow and denorm traps may or may not occur.
@end deftypefun

@deftypefun double mpf_get_d_2exp (signed long int *@var{exp}, const mpf_t @var{op})
Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding
towards zero), and with an exponent returned separately.

The return value is in the range @math{0.5@le{}@GMPabs{@var{d}}<1} and the
exponent is stored to @code{*@var{exp}}.  @m{@var{d} \times 2^{exp},
@var{d} * 2^@var{exp}} is the (truncated) @var{op} value.  If @var{op} is zero,
the return is @math{0.0} and 0 is stored to @code{*@var{exp}}.

@cindex @code{frexp}
This is similar to the standard C @code{frexp} function (@pxref{Normalization
Functions,,, libc, The GNU C Library Reference Manual}).
@end deftypefun

@deftypefun long mpf_get_si (const mpf_t @var{op})
@deftypefunx {unsigned long} mpf_get_ui (const mpf_t @var{op})
Convert @var{op} to a @code{long} or @code{unsigned long}, truncating any
fraction part.  If @var{op} is too big for the return type, the result is
undefined.

See also @code{mpf_fits_slong_p} and @code{mpf_fits_ulong_p}
(@pxref{Miscellaneous Float Functions}).
@end deftypefun

@deftypefun {char *} mpf_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{base}, size_t @var{n_digits}, const mpf_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  The base argument
may vary from 2 to 62 or from @minus{}2 to @minus{}36.  Up to @var{n_digits}
digits will be generated.  Trailing zeros are not returned.  No more digits
than can be accurately represented by @var{op} are ever generated.  If
@var{n_digits} is 0 then that accurate maximum number of digits are generated.

For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of
@math{@var{n_digits} + 2} bytes, that being enough for the mantissa, a
possible minus sign, and a null-terminator.  When @var{n_digits} is 0 to get
all significant digits, an application won't be able to know the space
required, and @var{str} should be @code{NULL} in that case.

The generated string is a fraction, with an implicit radix point immediately
to the left of the first digit.  The applicable exponent is written through
the @var{expptr} pointer.  For example, the number 3.1416 would be returned as
string @nicode{"31416"} and exponent 1.

When @var{op} is zero, an empty string is produced and the exponent returned
is 0.

A pointer to the result string is returned, being either the allocated block
or the given @var{str}.
@end deftypefun


@node Float Arithmetic, Float Comparison, Converting Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Float arithmetic functions
@cindex Arithmetic functions

@deftypefun void mpf_add (mpf_t @var{rop}, const mpf_t @var{op1}, const mpf_t @var{op2})
@deftypefunx void mpf_add_ui (mpf_t @var{rop}, const mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{op1} + @var{op2}}.
@end deftypefun

@deftypefun void mpf_sub (mpf_t @var{rop}, const mpf_t @var{op1}, const mpf_t @var{op2})
@deftypefunx void mpf_ui_sub (mpf_t @var{rop}, unsigned long int @var{op1}, const mpf_t @var{op2})
@deftypefunx void mpf_sub_ui (mpf_t @var{rop}, const mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @var{op1} @minus{} @var{op2}.
@end deftypefun

@deftypefun void mpf_mul (mpf_t @var{rop}, const mpf_t @var{op1}, const mpf_t @var{op2})
@deftypefunx void mpf_mul_ui (mpf_t @var{rop}, const mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

Division is undefined if the divisor is zero, and passing a zero divisor to the
divide functions will make these functions intentionally divide by zero.  This
lets the user handle arithmetic exceptions in these functions in the same
manner as other arithmetic exceptions.

@deftypefun void mpf_div (mpf_t @var{rop}, const mpf_t @var{op1}, const mpf_t @var{op2})
@deftypefunx void mpf_ui_div (mpf_t @var{rop}, unsigned long int @var{op1}, const mpf_t @var{op2})
@deftypefunx void mpf_div_ui (mpf_t @var{rop}, const mpf_t @var{op1}, unsigned long int @var{op2})
@cindex Division functions
Set @var{rop} to @var{op1}/@var{op2}.
@end deftypefun

@deftypefun void mpf_sqrt (mpf_t @var{rop}, const mpf_t @var{op})
@deftypefunx void mpf_sqrt_ui (mpf_t @var{rop}, unsigned long int @var{op})
@cindex Root extraction functions
Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}.
@end deftypefun

@deftypefun void mpf_pow_ui (mpf_t @var{rop}, const mpf_t @var{op1}, unsigned long int @var{op2})
@cindex Exponentiation functions
@cindex Powering functions
Set @var{rop} to @m{@var{op1}^{op2}, @var{op1} raised to the power @var{op2}}.
@end deftypefun

@deftypefun void mpf_neg (mpf_t @var{rop}, const mpf_t @var{op})
Set @var{rop} to @minus{}@var{op}.
@end deftypefun

@deftypefun void mpf_abs (mpf_t @var{rop}, const mpf_t @var{op})
Set @var{rop} to the absolute value of @var{op}.
@end deftypefun

@deftypefun void mpf_mul_2exp (mpf_t @var{rop}, const mpf_t @var{op1}, mp_bitcnt_t @var{op2})
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.
@end deftypefun

@deftypefun void mpf_div_2exp (mpf_t @var{rop}, const mpf_t @var{op1}, mp_bitcnt_t @var{op2})
Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
@var{op2}}.
@end deftypefun

@node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Float comparison functions
@cindex Comparison functions

@deftypefun int mpf_cmp (const mpf_t @var{op1}, const mpf_t @var{op2})
@deftypefunx int mpf_cmp_z (const mpf_t @var{op1}, const mpz_t @var{op2})
@deftypefunx int mpf_cmp_d (const mpf_t @var{op1}, double @var{op2})
@deftypefunx int mpf_cmp_ui (const mpf_t @var{op1}, unsigned long int @var{op2})
@deftypefunx int mpf_cmp_si (const mpf_t @var{op1}, signed long int @var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
@math{@var{op1} < @var{op2}}.

@code{mpf_cmp_d} can be called with an infinity, but results are undefined for
a NaN.
@end deftypefun

@deftypefun int mpf_eq (const mpf_t @var{op1}, const mpf_t @var{op2}, mp_bitcnt_t op3)
@strong{This function is mathematically ill-defined and should not be used.}

Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are
equal, zero otherwise.  Note that numbers like e.g., 256 (binary 100000000) and
255 (binary 11111111) will never be equal by this function's measure, and
furthermore that 0 will only be equal to itself.
@end deftypefun

@deftypefun void mpf_reldiff (mpf_t @var{rop}, const mpf_t @var{op1}, const mpf_t @var{op2})
Compute the relative difference between @var{op1} and @var{op2} and store the
result in @var{rop}.  This is @math{@GMPabs{@var{op1}-@var{op2}}/@var{op1}}.
@end deftypefun

@deftypefn Macro int mpf_sgn (const mpf_t @var{op})
@cindex Sign tests
@cindex Float sign tests
Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@var{op} < 0}.

This function is actually implemented as a macro.  It evaluates its argument
multiple times.
@end deftypefn

@node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Float input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

Functions that perform input from a stdio stream, and functions that output to
a stdio stream, of @code{mpf} numbers.  Passing a @code{NULL} pointer for a
@var{stream} argument to any of these functions will make them read from
@code{stdin} and write to @code{stdout}, respectively.

When using any of these functions, it is a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.

See also @ref{Formatted Output} and @ref{Formatted Input}.

@deftypefun size_t mpf_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n_digits}, const mpf_t @var{op})
Print @var{op} to @var{stream}, as a string of digits.  Return the number of
bytes written, or if an error occurred, return 0.

The mantissa is prefixed with an @samp{0.} and is in the given @var{base},
which may vary from 2 to 62 or from @minus{}2 to @minus{}36.  An exponent is
then printed, separated by an @samp{e}, or if the base is greater than 10 then
by an @samp{@@}.  The exponent is always in decimal.  The decimal point follows
the current locale, on systems providing @code{localeconv}.

For @var{base} in the range 2..36, digits and lower-case letters are used; for
@minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62,
digits, upper-case letters, and lower-case letters (in that significance order)
are used.

Up to @var{n_digits} will be printed from the mantissa, except that no more
digits than are accurately representable by @var{op} will be printed.
@var{n_digits} can be 0 to select that accurate maximum.
@end deftypefun

@deftypefun size_t mpf_inp_str (mpf_t @var{rop}, FILE *@var{stream}, int @var{base})
Read a string in base @var{base} from @var{stream}, and put the read float in
@var{rop}.  The string is of the form @samp{M@@N} or, if the base is 10 or
less, alternatively @samp{MeN}.  @samp{M} is the mantissa and @samp{N} is the
exponent.  The mantissa is always in the specified base.  The exponent is
either in the specified base or, if @var{base} is negative, in decimal.  The
decimal point expected is taken from the current locale, on systems providing
@code{localeconv}.

The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
@minus{}2.  Negative values are used to specify that the exponent is in
decimal.

Unlike the corresponding @code{mpz} function, the base will not be determined
from the leading characters of the string if @var{base} is 0.  This is so that
numbers like @samp{0.23} are not interpreted as octal.

Return the number of bytes read, or if an error occurred, return 0.
@end deftypefun

@c @deftypefun void mpf_out_raw (FILE *@var{stream}, const mpf_t @var{float})
@c Output @var{float} on stdio stream @var{stream}, in raw binary
@c format.  The float is written in a portable format, with 4 bytes of
@c size information, and that many bytes of limbs.  Both the size and the
@c limbs are written in decreasing significance order.
@c @end deftypefun

@c @deftypefun void mpf_inp_raw (mpf_t @var{float}, FILE *@var{stream})
@c Input from stdio stream @var{stream} in the format written by
@c @code{mpf_out_raw}, and put the result in @var{float}.
@c @end deftypefun


@node Miscellaneous Float Functions,  , I/O of Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous float functions
@cindex Float miscellaneous functions

@deftypefun void mpf_ceil (mpf_t @var{rop}, const mpf_t @var{op})
@deftypefunx void mpf_floor (mpf_t @var{rop}, const mpf_t @var{op})
@deftypefunx void mpf_trunc (mpf_t @var{rop}, const mpf_t @var{op})
@cindex Rounding functions
@cindex Float rounding functions
Set @var{rop} to @var{op} rounded to an integer.  @code{mpf_ceil} rounds to the
next higher integer, @code{mpf_floor} to the next lower, and @code{mpf_trunc}
to the integer towards zero.
@end deftypefun

@deftypefun int mpf_integer_p (const mpf_t @var{op})
Return non-zero if @var{op} is an integer.
@end deftypefun

@deftypefun int mpf_fits_ulong_p (const mpf_t @var{op})
@deftypefunx int mpf_fits_slong_p (const mpf_t @var{op})
@deftypefunx int mpf_fits_uint_p (const mpf_t @var{op})
@deftypefunx int mpf_fits_sint_p (const mpf_t @var{op})
@deftypefunx int mpf_fits_ushort_p (const mpf_t @var{op})
@deftypefunx int mpf_fits_sshort_p (const mpf_t @var{op})
Return non-zero if @var{op} would fit in the respective C data type, when
truncated to an integer.
@end deftypefun

@deftypefun void mpf_urandomb (mpf_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{nbits})
@cindex Random number functions
@cindex Float random number functions
Generate a uniformly distributed random float in @var{rop}, such that @math{0
@le{} @var{rop} < 1}, with @var{nbits} significant bits in the mantissa or
less if the precision of @var{rop} is smaller.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization}) before
invoking this function.
@end deftypefun

@deftypefun void mpf_random2 (mpf_t @var{rop}, mp_size_t @var{max_size}, mp_exp_t @var{exp})
Generate a random float of at most @var{max_size} limbs, with long strings of
zeros and ones in the binary representation.  The exponent of the number is in
the interval @minus{}@var{exp} to @var{exp} (in limbs).  This function is
useful for testing functions and algorithms, since these kind of random
numbers have proven to be more likely to trigger corner-case bugs.  Negative
random numbers are generated when @var{max_size} is negative.
@end deftypefun

@c @deftypefun size_t mpf_size (const mpf_t @var{op})
@c Return the size of @var{op} measured in number of limbs.  If @var{op} is
@c zero, the returned value will be zero.  (@xref{Nomenclature}, for an
@c explanation of the concept @dfn{limb}.)
@c
@c @strong{This function is obsolete.  It will disappear from future GMP
@c releases.}
@c @end deftypefun


@node Low-level Functions, Random Number Functions, Floating-point Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Low-level Functions
@cindex Low-level functions

This chapter describes low-level GMP functions, used to implement the
high-level GMP functions, but also intended for time-critical user code.

These functions start with the prefix @code{mpn_}.

@c 1. Some of these function clobber input operands.
@c

The @code{mpn} functions are designed to be as fast as possible, @strong{not}
to provide a coherent calling interface.  The different functions have somewhat
similar interfaces, but there are variations that make them hard to use.  These
functions do as little as possible apart from the real multiple precision
computation, so that no time is spent on things that not all callers need.

A source operand is specified by a pointer to the least significant limb and a
limb count.  A destination operand is specified by just a pointer.  It is the
responsibility of the caller to ensure that the destination has enough space
for storing the result.

With this way of specifying operands, it is possible to perform computations on
subranges of an argument, and store the result into a subrange of a
destination.

A common requirement for all functions is that each source area needs at least
one limb.  No size argument may be zero.  Unless otherwise stated, in-place
operations are allowed where source and destination are the same, but not where
they only partly overlap.

The @code{mpn} functions are the base for the implementation of the
@code{mpz_}, @code{mpf_}, and @code{mpq_} functions.

This example adds the number beginning at @var{s1p} and the number beginning at
@var{s2p} and writes the sum at @var{destp}.  All areas have @var{n} limbs.

@example
cy = mpn_add_n (destp, s1p, s2p, n)
@end example

It should be noted that the @code{mpn} functions make no attempt to identify
high or low zero limbs on their operands, or other special forms.  On random
data such cases will be unlikely and it'd be wasteful for every function to
check every time.  An application knowing something about its data can take
steps to trim or perhaps split its calculations.
@c
@c  For reference, within gmp mpz_t operands never have high zero limbs, and
@c  we rate low zero limbs as unlikely too (or something an application should
@c  handle).  This is a prime motivation for not stripping zero limbs in say
@c  mpn_mul_n etc.
@c
@c  Other applications doing variable-length calculations will quite likely do
@c  something similar to mpz.  And even if not then it's highly likely zero
@c  limb stripping can be done at just a few judicious points, which will be
@c  more efficient than having lots of mpn functions checking every time.

@sp 1
@noindent
In the notation used below, a source operand is identified by the pointer to
the least significant limb, and the limb count in braces.  For example,
@{@var{s1p}, @var{s1n}@}.

@deftypefun mp_limb_t mpn_add_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Add @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the @var{n}
least significant limbs of the result to @var{rp}.  Return carry, either 0 or
1.

This is the lowest-level function for addition.  It is the preferred function
for addition, since it is written in assembly for most CPUs.  For addition of
a variable to itself (i.e., @var{s1p} equals @var{s2p}) use @code{mpn_lshift}
with a count of 1 for optimal speed.
@end deftypefun

@deftypefun mp_limb_t mpn_add_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Add @{@var{s1p}, @var{n}@} and @var{s2limb}, and write the @var{n} least
significant limbs of the result to @var{rp}.  Return carry, either 0 or 1.
@end deftypefun

@deftypefun mp_limb_t mpn_add (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Add @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
@var{s1n} least significant limbs of the result to @var{rp}.  Return carry,
either 0 or 1.

This function requires that @var{s1n} is greater than or equal to @var{s2n}.
@end deftypefun

@deftypefun mp_limb_t mpn_sub_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Subtract @{@var{s2p}, @var{n}@} from @{@var{s1p}, @var{n}@}, and write the
@var{n} least significant limbs of the result to @var{rp}.  Return borrow,
either 0 or 1.

This is the lowest-level function for subtraction.  It is the preferred
function for subtraction, since it is written in assembly for most CPUs.
@end deftypefun

@deftypefun mp_limb_t mpn_sub_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Subtract @var{s2limb} from @{@var{s1p}, @var{n}@}, and write the @var{n} least
significant limbs of the result to @var{rp}.  Return borrow, either 0 or 1.
@end deftypefun

@deftypefun mp_limb_t mpn_sub (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Subtract @{@var{s2p}, @var{s2n}@} from @{@var{s1p}, @var{s1n}@}, and write the
@var{s1n} least significant limbs of the result to @var{rp}.  Return borrow,
either 0 or 1.

This function requires that @var{s1n} is greater than or equal to
@var{s2n}.
@end deftypefun

@deftypefun mp_limb_t mpn_neg (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
Perform the negation of @{@var{sp}, @var{n}@}, and write the result to
@{@var{rp}, @var{n}@}.  This is equivalent to calling @code{mpn_sub_n} with a
@var{n}-limb zero minuend and passing @{@var{sp}, @var{n}@} as subtrahend.
Return borrow, either 0 or 1.
@end deftypefun

@deftypefun void mpn_mul_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Multiply @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the
2*@var{n}-limb result to @var{rp}.

The destination has to have space for 2*@var{n} limbs, even if the product's
most significant limb is zero.  No overlap is permitted between the
destination and either source.

If the two input operands are the same, use @code{mpn_sqr}.
@end deftypefun

@deftypefun mp_limb_t mpn_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Multiply @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
(@var{s1n}+@var{s2n})-limb result to @var{rp}.  Return the most significant
limb of the result.

The destination has to have space for @var{s1n} + @var{s2n} limbs, even if the
product's most significant limb is zero.  No overlap is permitted between the
destination and either source.

This function requires that @var{s1n} is greater than or equal to @var{s2n}.
@end deftypefun

@deftypefun void mpn_sqr (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n})
Compute the square of @{@var{s1p}, @var{n}@} and write the 2*@var{n}-limb
result to @var{rp}.

The destination has to have space for 2@var{n} limbs, even if the result's
most significant limb is zero.  No overlap is permitted between the
destination and the source.
@end deftypefun

@deftypefun mp_limb_t mpn_mul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{n}@} by @var{s2limb}, and write the @var{n} least
significant limbs of the product to @var{rp}.  Return the most significant
limb of the product.  @{@var{s1p}, @var{n}@} and @{@var{rp}, @var{n}@} are
allowed to overlap provided @math{@var{rp} @le{} @var{s1p}}.

This is a low-level function that is a building block for general
multiplication as well as other operations in GMP@.  It is written in assembly
for most CPUs.

Don't call this function if @var{s2limb} is a power of 2; use @code{mpn_lshift}
with a count equal to the logarithm of @var{s2limb} instead, for optimal speed.
@end deftypefun

@deftypefun mp_limb_t mpn_addmul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and add the @var{n} least
significant limbs of the product to @{@var{rp}, @var{n}@} and write the result
to @var{rp}.  Return the most significant limb of the product, plus carry-out
from the addition.  @{@var{s1p}, @var{n}@} and @{@var{rp}, @var{n}@} are
allowed to overlap provided @math{@var{rp} @le{} @var{s1p}}.

This is a low-level function that is a building block for general
multiplication as well as other operations in GMP@.  It is written in assembly
for most CPUs.
@end deftypefun

@deftypefun mp_limb_t mpn_submul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and subtract the @var{n}
least significant limbs of the product from @{@var{rp}, @var{n}@} and write the
result to @var{rp}.  Return the most significant limb of the product, plus
borrow-out from the subtraction.  @{@var{s1p}, @var{n}@} and @{@var{rp},
@var{n}@} are allowed to overlap provided @math{@var{rp} @le{} @var{s1p}}.

This is a low-level function that is a building block for general
multiplication and division as well as other operations in GMP@.  It is written
in assembly for most CPUs.
@end deftypefun

@deftypefun void mpn_tdiv_qr (mp_limb_t *@var{qp}, mp_limb_t *@var{rp}, mp_size_t @var{qxn}, const mp_limb_t *@var{np}, mp_size_t @var{nn}, const mp_limb_t *@var{dp}, mp_size_t @var{dn})
Divide @{@var{np}, @var{nn}@} by @{@var{dp}, @var{dn}@} and put the quotient
at @{@var{qp}, @var{nn}@minus{}@var{dn}+1@} and the remainder at @{@var{rp},
@var{dn}@}.  The quotient is rounded towards 0.

No overlap is permitted between arguments, except that @var{np} might equal
@var{rp}.  The dividend size @var{nn} must be greater than or equal to divisor
size @var{dn}.  The most significant limb of the divisor must be non-zero.  The
@var{qxn} operand must be zero.
@end deftypefun

@deftypefun mp_limb_t mpn_divrem (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
[This function is obsolete.  Please call @code{mpn_tdiv_qr} instead for best
performance.]

Divide @{@var{rs2p}, @var{rs2n}@} by @{@var{s3p}, @var{s3n}@}, and write the
quotient at @var{r1p}, with the exception of the most significant limb, which
is returned.  The remainder replaces the dividend at @var{rs2p}; it will be
@var{s3n} limbs long (i.e., as many limbs as the divisor).

In addition to an integer quotient, @var{qxn} fraction limbs are developed, and
stored after the integral limbs.  For most usages, @var{qxn} will be zero.

It is required that @var{rs2n} is greater than or equal to @var{s3n}.  It is
required that the most significant bit of the divisor is set.

If the quotient is not needed, pass @var{rs2p} + @var{s3n} as @var{r1p}.  Aside
from that special case, no overlap between arguments is permitted.

Return the most significant limb of the quotient, either 0 or 1.

The area at @var{r1p} needs to be @var{rs2n} @minus{} @var{s3n} + @var{qxn}
limbs large.
@end deftypefun

@deftypefn Function mp_limb_t mpn_divrem_1 (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, @w{mp_limb_t *@var{s2p}}, mp_size_t @var{s2n}, mp_limb_t @var{s3limb})
@deftypefnx Macro mp_limb_t mpn_divmod_1 (mp_limb_t *@var{r1p}, mp_limb_t *@var{s2p}, @w{mp_size_t @var{s2n}}, @w{mp_limb_t @var{s3limb}})
Divide @{@var{s2p}, @var{s2n}@} by @var{s3limb}, and write the quotient at
@var{r1p}.  Return the remainder.

The integer quotient is written to @{@var{r1p}+@var{qxn}, @var{s2n}@} and in
addition @var{qxn} fraction limbs are developed and written to @{@var{r1p},
@var{qxn}@}.  Either or both @var{s2n} and @var{qxn} can be zero.  For most
usages, @var{qxn} will be zero.

@code{mpn_divmod_1} exists for upward source compatibility and is simply a
macro calling @code{mpn_divrem_1} with a @var{qxn} of 0.

The areas at @var{r1p} and @var{s2p} have to be identical or completely
separate, not partially overlapping.
@end deftypefn

@deftypefun mp_limb_t mpn_divmod (mp_limb_t *@var{r1p}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
[This function is obsolete.  Please call @code{mpn_tdiv_qr} instead for best
performance.]
@end deftypefun

@deftypefun void mpn_divexact_1 (mp_limb_t * @var{rp}, const mp_limb_t * @var{sp}, mp_size_t @var{n}, mp_limb_t @var{d})
Divide @{@var{sp}, @var{n}@} by @var{d}, expecting it to divide exactly, and
writing the result to @{@var{rp}, @var{n}@}. If @var{d} doesn't divide
exactly, the value written to @{@var{rp}, @var{n}@} is undefined. The areas at
@var{rp} and @var{sp} have to be identical or completely separate, not
partially overlapping.
@end deftypefun

@deftypefn Macro mp_limb_t mpn_divexact_by3 (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}})
@deftypefnx Function mp_limb_t mpn_divexact_by3c (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}}, mp_limb_t @var{carry})
Divide @{@var{sp}, @var{n}@} by 3, expecting it to divide exactly, and writing
the result to @{@var{rp}, @var{n}@}.  If 3 divides exactly, the return value is
zero and the result is the quotient.  If not, the return value is non-zero and
the result won't be anything useful.

@code{mpn_divexact_by3c} takes an initial carry parameter, which can be the
return value from a previous call, so a large calculation can be done piece by
piece from low to high.  @code{mpn_divexact_by3} is simply a macro calling
@code{mpn_divexact_by3c} with a 0 carry parameter.

These routines use a multiply-by-inverse and will be faster than
@code{mpn_divrem_1} on CPUs with fast multiplication but slow division.

The source @math{a}, result @math{q}, size @math{n}, initial carry @math{i},
and return value @math{c} satisfy @m{cb^n+a-i=3q, c*b^n + a-i = 3*q}, where
@m{b=2\GMPraise{@code{GMP\_NUMB\_BITS}}, b=2^GMP_NUMB_BITS}.  The
return @math{c} is always 0, 1 or 2, and the initial carry @math{i} must also
be 0, 1 or 2 (these are both borrows really).  When @math{c=0} clearly
@math{q=(a-i)/3}.  When @m{c \neq 0, c!=0}, the remainder @math{(a-i) @bmod{}
3} is given by @math{3-c}, because @math{b @equiv{} 1 @bmod{} 3} (when
@code{mp_bits_per_limb} is even, which is always so currently).
@end deftypefn

@deftypefun mp_limb_t mpn_mod_1 (const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
Divide @{@var{s1p}, @var{s1n}@} by @var{s2limb}, and return the remainder.
@var{s1n} can be zero.
@end deftypefun

@deftypefun mp_limb_t mpn_lshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
Shift @{@var{sp}, @var{n}@} left by @var{count} bits, and write the result to
@{@var{rp}, @var{n}@}.  The bits shifted out at the left are returned in the
least significant @var{count} bits of the return value (the rest of the return
value is zero).

@var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1.  The
regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
@math{@var{rp} @ge{} @var{sp}}.

This function is written in assembly for most CPUs.
@end deftypefun

@deftypefun mp_limb_t mpn_rshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
Shift @{@var{sp}, @var{n}@} right by @var{count} bits, and write the result to
@{@var{rp}, @var{n}@}.  The bits shifted out at the right are returned in the
most significant @var{count} bits of the return value (the rest of the return
value is zero).

@var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1.  The
regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
@math{@var{rp} @le{} @var{sp}}.

This function is written in assembly for most CPUs.
@end deftypefun

@deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Compare @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@} and return a
positive value if @math{@var{s1} > @var{s2}}, 0 if they are equal, or a
negative value if @math{@var{s1} < @var{s2}}.
@end deftypefun

@deftypefun int mpn_zero_p (const mp_limb_t *@var{sp}, mp_size_t @var{n})
Test @{@var{sp}, @var{n}@} and return 1 if the operand is zero, 0 otherwise.
@end deftypefun

@deftypefun mp_size_t mpn_gcd (mp_limb_t *@var{rp}, mp_limb_t *@var{xp}, mp_size_t @var{xn}, mp_limb_t *@var{yp}, mp_size_t @var{yn})
Set @{@var{rp}, @var{retval}@} to the greatest common divisor of @{@var{xp},
@var{xn}@} and @{@var{yp}, @var{yn}@}.  The result can be up to @var{yn} limbs,
the return value is the actual number produced.  Both source operands are
destroyed.

It is required that @math{@var{xn} @ge @var{yn} > 0}, the most significant
limb of @{@var{yp}, @var{yn}@} must be non-zero, and at least one of
the two operands must be odd.  No overlap is permitted
between @{@var{xp}, @var{xn}@} and @{@var{yp}, @var{yn}@}.
@end deftypefun

@deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t *@var{xp}, mp_size_t @var{xn}, mp_limb_t @var{ylimb})
Return the greatest common divisor of @{@var{xp}, @var{xn}@} and @var{ylimb}.
Both operands must be non-zero.
@end deftypefun

@deftypefun mp_size_t mpn_gcdext (mp_limb_t *@var{gp}, mp_limb_t *@var{sp}, mp_size_t *@var{sn}, mp_limb_t *@var{up}, mp_size_t @var{un}, mp_limb_t *@var{vp}, mp_size_t @var{vn})
Let @m{U,@var{U}} be defined by @{@var{up}, @var{un}@} and let @m{V,@var{V}} be
defined by @{@var{vp}, @var{vn}@}.

Compute the greatest common divisor @math{G} of @math{U} and @math{V}.  Compute
a cofactor @math{S} such that @math{G = US + VT}.  The second cofactor @var{T}
is not computed but can easily be obtained from @m{(G - US) / V, (@var{G} -
@var{U}*@var{S}) / @var{V}} (the division will be exact).  It is required that
@math{@var{un} @ge @var{vn} > 0}, and the most significant
limb of @{@var{vp}, @var{vn}@} must be non-zero.

@math{S} satisfies @math{S = 1} or @math{@GMPabs{S} < V / (2 G)}. @math{S =
0} if and only if @math{V} divides @math{U} (i.e., @math{G = V}).

Store @math{G} at @var{gp} and let the return value define its limb count.
Store @math{S} at @var{sp} and let |*@var{sn}| define its limb count.  @math{S}
can be negative; when this happens *@var{sn} will be negative.  The area at
@var{gp} should have room for @var{vn} limbs and the area at @var{sp} should
have room for @math{@var{vn}+1} limbs.

Both source operands are destroyed.

Compatibility notes: GMP 4.3.0 and 4.3.1 defined @math{S} less strictly.
Earlier as well as later GMP releases define @math{S} as described here.
GMP releases before GMP 4.3.0 required additional space for both input and output
areas. More precisely, the areas @{@var{up}, @math{@var{un}+1}@} and
@{@var{vp}, @math{@var{vn}+1}@} were destroyed (i.e.@: the operands plus an
extra limb past the end of each), and the areas pointed to by @var{gp} and
@var{sp} should each have room for @math{@var{un}+1} limbs.
@end deftypefun

@deftypefun mp_size_t mpn_sqrtrem (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
Compute the square root of @{@var{sp}, @var{n}@} and put the result at
@{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and the remainder at @{@var{r2p},
@var{retval}@}.  @var{r2p} needs space for @var{n} limbs, but the return value
indicates how many are produced.

The most significant limb of @{@var{sp}, @var{n}@} must be non-zero.  The
areas @{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and @{@var{sp}, @var{n}@} must
be completely separate.  The areas @{@var{r2p}, @var{n}@} and @{@var{sp},
@var{n}@} must be either identical or completely separate.

If the remainder is not wanted then @var{r2p} can be @code{NULL}, and in this
case the return value is zero or non-zero according to whether the remainder
would have been zero or non-zero.

A return value of zero indicates a perfect square.  See also
@code{mpn_perfect_square_p}.
@end deftypefun

@deftypefun size_t mpn_sizeinbase (const mp_limb_t *@var{xp}, mp_size_t @var{n}, int @var{base})
Return the size of @{@var{xp},@var{n}@} measured in number of digits in the
given @var{base}.  @var{base} can vary from 2 to 62.  Requires @math{@var{n} > 0}
and @math{@var{xp}[@var{n}-1] > 0}.  The result will be either exact or
1 too big.  If @var{base} is a power of 2, the result is always exact.
@end deftypefun

@deftypefun mp_size_t mpn_get_str (unsigned char *@var{str}, int @var{base}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n})
Convert @{@var{s1p}, @var{s1n}@} to a raw unsigned char array at @var{str} in
base @var{base}, and return the number of characters produced.  There may be
leading zeros in the string.  The string is not in ASCII; to convert it to
printable format, add the ASCII codes for @samp{0} or @samp{A}, depending on
the base and range.  @var{base} can vary from 2 to 256.

The most significant limb of the input @{@var{s1p}, @var{s1n}@} must be
non-zero.  The input @{@var{s1p}, @var{s1n}@} is clobbered, except when
@var{base} is a power of 2, in which case it's unchanged.

The area at @var{str} has to have space for the largest possible number
represented by a @var{s1n} long limb array, plus one extra character.
@end deftypefun

@deftypefun mp_size_t mpn_set_str (mp_limb_t *@var{rp}, const unsigned char *@var{str}, size_t @var{strsize}, int @var{base})
Convert bytes @{@var{str},@var{strsize}@} in the given @var{base} to limbs at
@var{rp}.

@math{@var{str}[0]} is the most significant input byte and
@math{@var{str}[@var{strsize}-1]} is the least significant input byte.  Each
byte should be a value in the range 0 to @math{@var{base}-1}, not an ASCII
character.  @var{base} can vary from 2 to 256.

The converted value is @{@var{rp},@var{rn}@} where @var{rn} is the return
value.  If the most significant input byte @math{@var{str}[0]} is non-zero,
then @math{@var{rp}[@var{rn}-1]} will be non-zero, else
@math{@var{rp}[@var{rn}-1]} and some number of subsequent limbs may be zero.

The area at @var{rp} has to have space for the largest possible number with
@var{strsize} digits in the chosen base, plus one extra limb.

The input must have at least one byte, and no overlap is permitted between
@{@var{str},@var{strsize}@} and the result at @var{rp}.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpn_scan0 (const mp_limb_t *@var{s1p}, mp_bitcnt_t @var{bit})
Scan @var{s1p} from bit position @var{bit} for the next clear bit.

It is required that there be a clear bit within the area at @var{s1p} at or
beyond bit position @var{bit}, so that the function has something to return.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpn_scan1 (const mp_limb_t *@var{s1p}, mp_bitcnt_t @var{bit})
Scan @var{s1p} from bit position @var{bit} for the next set bit.

It is required that there be a set bit within the area at @var{s1p} at or
beyond bit position @var{bit}, so that the function has something to return.
@end deftypefun

@deftypefun void mpn_random (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
@deftypefunx void mpn_random2 (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
Generate a random number of length @var{r1n} and store it at @var{r1p}.  The
most significant limb is always non-zero.  @code{mpn_random} generates
uniformly distributed limb data, @code{mpn_random2} generates long strings of
zeros and ones in the binary representation.

@code{mpn_random2} is intended for testing the correctness of the @code{mpn}
routines.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpn_popcount (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
Count the number of set bits in @{@var{s1p}, @var{n}@}.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpn_hamdist (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Compute the hamming distance between @{@var{s1p}, @var{n}@} and @{@var{s2p},
@var{n}@}, which is the number of bit positions where the two operands have
different bit values.
@end deftypefun

@deftypefun int mpn_perfect_square_p (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
Return non-zero iff @{@var{s1p}, @var{n}@} is a perfect square.
The most significant limb of the input @{@var{s1p}, @var{n}@} must be
non-zero.
@end deftypefun

@deftypefun void mpn_and_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and @{@var{s2p},
@var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_ior_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and
@{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_xor_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Perform the bitwise logical exclusive or of @{@var{s1p}, @var{n}@} and
@{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_andn_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and the bitwise
complement of @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_iorn_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and the bitwise
complement of @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_nand_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and @{@var{s2p},
@var{n}@}, and write the bitwise complement of the result to @{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_nior_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and
@{@var{s2p}, @var{n}@}, and write the bitwise complement of the result to
@{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_xnor_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Perform the bitwise logical exclusive or of @{@var{s1p}, @var{n}@} and
@{@var{s2p}, @var{n}@}, and write the bitwise complement of the result to
@{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_com (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
Perform the bitwise complement of @{@var{sp}, @var{n}@}, and write the result
to @{@var{rp}, @var{n}@}.
@end deftypefun

@deftypefun void mpn_copyi (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n})
Copy from @{@var{s1p}, @var{n}@} to @{@var{rp}, @var{n}@}, increasingly.
@end deftypefun

@deftypefun void mpn_copyd (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n})
Copy from @{@var{s1p}, @var{n}@} to @{@var{rp}, @var{n}@}, decreasingly.
@end deftypefun

@deftypefun void mpn_zero (mp_limb_t *@var{rp}, mp_size_t @var{n})
Zero @{@var{rp}, @var{n}@}.
@end deftypefun

@sp 1
@section Low-level functions for cryptography
@cindex Low-level functions for cryptography
@cindex Cryptography functions, low-level

The functions prefixed with @code{mpn_sec_} and @code{mpn_cnd_} are designed to
perform the exact same low-level operations and have the same cache access
patterns for any two same-size arguments, assuming that function arguments are
placed at the same position and that the machine state is identical upon
function entry.  These functions are intended for cryptographic purposes, where
resilience to side-channel attacks is desired.

These functions are less efficient than their ``leaky'' counterparts; their
performance for operands of the sizes typically used for cryptographic
applications is between 15% and 100% worse.  For larger operands, these
functions might be inadequate, since they rely on asymptotically elementary
algorithms.

These functions do not make any explicit allocations.  Those of these functions
that need scratch space accept a scratch space operand.  This convention allows
callers to keep sensitive data in designated memory areas.  Note however that
compilers may choose to spill scalar values used within these functions to
their stack frame and that such scalars may contain sensitive data.

In addition to these specially crafted functions, the following @code{mpn}
functions are naturally side-channel resistant: @code{mpn_add_n},
@code{mpn_sub_n}, @code{mpn_lshift}, @code{mpn_rshift}, @code{mpn_zero},
@code{mpn_copyi}, @code{mpn_copyd}, @code{mpn_com}, and the logical function
(@code{mpn_and_n}, etc).

There are some exceptions from the side-channel resilience: (1) Some assembly
implementations of @code{mpn_lshift} identify shift-by-one as a special case.
This is a problem iff the shift count is a function of sensitive data.  (2)
Alpha ev6 and Pentium4 using 64-bit limbs have leaky @code{mpn_add_n} and
@code{mpn_sub_n}.  (3) Alpha ev6 has a leaky @code{mpn_mul_1} which also makes
@code{mpn_sec_mul} on those systems unsafe.

@deftypefun mp_limb_t mpn_cnd_add_n (mp_limb_t @var{cnd}, mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
@deftypefunx mp_limb_t mpn_cnd_sub_n (mp_limb_t @var{cnd}, mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
These functions do conditional addition and subtraction.  If @var{cnd} is
non-zero, they produce the same result as a regular @code{mpn_add_n} or
@code{mpn_sub_n}, and if @var{cnd} is zero, they copy @{@var{s1p},@var{n}@} to
the result area and return zero.  The functions are designed to have timing and
memory access patterns depending only on size and location of the data areas,
but independent of the condition @var{cnd}.  Like for @code{mpn_add_n} and
@code{mpn_sub_n}, on most machines, the timing will also be independent of the
actual limb values.
@end deftypefun

@deftypefun mp_limb_t mpn_sec_add_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{ap}, mp_size_t @var{n}, mp_limb_t @var{b}, mp_limb_t *@var{tp})
@deftypefunx mp_limb_t mpn_sec_sub_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{ap}, mp_size_t @var{n}, mp_limb_t @var{b}, mp_limb_t *@var{tp})
Set @var{R} to @var{A} + @var{b} or @var{A} - @var{b}, respectively, where
@var{R} = @{@var{rp},@var{n}@}, @var{A} = @{@var{ap},@var{n}@}, and @var{b} is
a single limb. Returns carry.

These functions take @math{O(N)} time, unlike the leaky functions
@code{mpn_add_1} which are @math{O(1)} on average. They require scratch space
of @code{mpn_sec_add_1_itch(@var{n})} and @code{mpn_sec_sub_1_itch(@var{n})}
limbs, respectively, to be passed in the @var{tp} parameter. The scratch space
requirements are guaranteed to be at most @var{n} limbs, and increase
monotonously in the operand size.
@end deftypefun

@deftypefun void mpn_cnd_swap (mp_limb_t @var{cnd}, volatile mp_limb_t *@var{ap}, volatile mp_limb_t *@var{bp}, mp_size_t @var{n})
If @var{cnd} is non-zero, swaps the contents of the areas @{@var{ap},@var{n}@}
and @{@var{bp},@var{n}@}. Otherwise, the areas are left unmodified.
Implemented using logical operations on the limbs, with the same memory
accesses independent of the value of @var{cnd}.
@end deftypefun

@deftypefun void mpn_sec_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{ap}, mp_size_t @var{an}, const mp_limb_t *@var{bp}, mp_size_t @var{bn}, mp_limb_t *@var{tp})
@deftypefunx mp_size_t mpn_sec_mul_itch (mp_size_t @var{an}, mp_size_t @var{bn})
Set @var{R} to @math{A @times{} B}, where @var{A} = @{@var{ap},@var{an}@},
@var{B} = @{@var{bp},@var{bn}@}, and @var{R} =
@{@var{rp},@math{@var{an}+@var{bn}}@}.

It is required that @math{@var{an} @ge @var{bn} > 0}.

No overlapping between @var{R} and the input operands is allowed.  For
@math{@var{A} = @var{B}}, use @code{mpn_sec_sqr} for optimal performance.

This function requires scratch space of @code{mpn_sec_mul_itch(@var{an},
@var{bn})} limbs to be passed in the @var{tp} parameter.  The scratch space
requirements are guaranteed to increase monotonously in the operand sizes.
@end deftypefun


@deftypefun void mpn_sec_sqr (mp_limb_t *@var{rp}, const mp_limb_t *@var{ap}, mp_size_t @var{an}, mp_limb_t *@var{tp})
@deftypefunx mp_size_t mpn_sec_sqr_itch (mp_size_t @var{an})
Set @var{R} to @math{A^2}, where @var{A} = @{@var{ap},@var{an}@}, and @var{R} =
@{@var{rp},@math{2@var{an}}@}.

It is required that @math{@var{an} > 0}.

No overlapping between @var{R} and the input operands is allowed.

This function requires scratch space of @code{mpn_sec_sqr_itch(@var{an})} limbs
to be passed in the @var{tp} parameter.  The scratch space requirements are
guaranteed to increase monotonously in the operand size.
@end deftypefun


@deftypefun void mpn_sec_powm (mp_limb_t *@var{rp}, const mp_limb_t *@var{bp}, mp_size_t @var{bn}, const mp_limb_t *@var{ep}, mp_bitcnt_t @var{enb},  const mp_limb_t *@var{mp}, mp_size_t @var{n}, mp_limb_t *@var{tp})
@deftypefunx mp_size_t mpn_sec_powm_itch (mp_size_t @var{bn}, mp_bitcnt_t @var{enb}, size_t @var{n})
Set @var{R} to @m{B^E \bmod @var{M}, (@var{B} raised to @var{E}) modulo
@var{M}}, where @var{R} = @{@var{rp},@var{n}@}, @var{M} = @{@var{mp},@var{n}@},
and @var{E} = @{@var{ep},@math{@GMPceil{@var{enb} /
@code{GMP\_NUMB\_BITS}}}@}.

It is required that @math{@var{B} > 0}, that @math{@var{M} > 0} is odd, and
that @m{@var{E} < 2@GMPraise{@var{enb}}, @var{E} < 2^@var{enb}}, with @math{@var{enb} > 0}.

No overlapping between @var{R} and the input operands is allowed.

This function requires scratch space of @code{mpn_sec_powm_itch(@var{bn},
@var{enb}, @var{n})} limbs to be passed in the @var{tp} parameter.  The scratch
space requirements are guaranteed to increase monotonously in the operand
sizes.
@end deftypefun

@deftypefun void mpn_sec_tabselect (mp_limb_t *@var{rp}, const mp_limb_t *@var{tab}, mp_size_t @var{n}, mp_size_t @var{nents}, mp_size_t @var{which})
Select entry @var{which} from table @var{tab}, which has @var{nents} entries, each @var{n}
limbs.  Store the selected entry at @var{rp}.

This function reads the entire table to avoid side-channel information leaks.
@end deftypefun

@deftypefun mp_limb_t mpn_sec_div_qr (mp_limb_t *@var{qp}, mp_limb_t *@var{np}, mp_size_t @var{nn}, const mp_limb_t *@var{dp}, mp_size_t @var{dn}, mp_limb_t *@var{tp})
@deftypefunx mp_size_t mpn_sec_div_qr_itch (mp_size_t @var{nn}, mp_size_t @var{dn})

Set @var{Q} to @m{\lfloor @var{N} / @var{D}\rfloor, the truncated quotient
@var{N} / @var{D}} and @var{R} to @m{@var{N} \bmod @var{D}, @var{N} modulo
@var{D}}, where @var{N} = @{@var{np},@var{nn}@}, @var{D} =
@{@var{dp},@var{dn}@}, @var{Q}'s most significant limb is the function return
value and the remaining limbs are @{@var{qp},@var{nn-dn}@}, and @var{R} =
@{@var{np},@var{dn}@}.

It is required that @math{@var{nn} @ge @var{dn} @ge 1}, and that
@m{@var{dp}[@var{dn}-1] @neq 0, @var{dp}[@var{dn}-1] != 0}.  This does not
imply that @math{@var{N} @ge @var{D}} since @var{N} might be zero-padded.

Note the overlapping between @var{N} and @var{R}.  No other operand overlapping
is allowed.  The entire space occupied by @var{N} is overwritten.

This function requires scratch space of @code{mpn_sec_div_qr_itch(@var{nn},
@var{dn})} limbs to be passed in the @var{tp} parameter.
@end deftypefun

@deftypefun void mpn_sec_div_r (mp_limb_t *@var{np}, mp_size_t @var{nn}, const mp_limb_t *@var{dp}, mp_size_t @var{dn}, mp_limb_t *@var{tp})
@deftypefunx mp_size_t mpn_sec_div_r_itch (mp_size_t @var{nn}, mp_size_t @var{dn})

Set @var{R} to @m{@var{N} \bmod @var{D}, @var{N} modulo @var{D}}, where @var{N}
= @{@var{np},@var{nn}@}, @var{D} = @{@var{dp},@var{dn}@}, and @var{R} =
@{@var{np},@var{dn}@}.

It is required that @math{@var{nn} @ge @var{dn} @ge 1}, and that
@m{@var{dp}[@var{dn}-1] @neq 0, @var{dp}[@var{dn}-1] != 0}.  This does not
imply that @math{@var{N} @ge @var{D}} since @var{N} might be zero-padded.

Note the overlapping between @var{N} and @var{R}.  No other operand overlapping
is allowed.  The entire space occupied by @var{N} is overwritten.

This function requires scratch space of @code{mpn_sec_div_r_itch(@var{nn},
@var{dn})} limbs to be passed in the @var{tp} parameter.
@end deftypefun

@deftypefun int mpn_sec_invert (mp_limb_t *@var{rp}, mp_limb_t *@var{ap}, const mp_limb_t *@var{mp}, mp_size_t @var{n}, mp_bitcnt_t @var{nbcnt}, mp_limb_t *@var{tp})
@deftypefunx mp_size_t mpn_sec_invert_itch (mp_size_t @var{n})
Set @var{R} to @m{@var{A}^{-1} \bmod @var{M}, the inverse of @var{A} modulo
@var{M}}, where @var{R} = @{@var{rp},@var{n}@}, @var{A} = @{@var{ap},@var{n}@},
and @var{M} = @{@var{mp},@var{n}@}.  @strong{This function's interface is
preliminary.}

If an inverse exists, return 1, otherwise return 0 and leave @var{R}
undefined. In either case, the input @var{A} is destroyed.

It is required that @var{M} is odd, and that @math{@var{nbcnt} @ge
@GMPceil{\log(@var{A}+1)} + @GMPceil{\log(@var{M}+1)}}.  A safe choice is
@m{@var{nbcnt} = 2@var{n} @times{} @code{GMP\_NUMB\_BITS}, @var{nbcnt} = 2
@times{} @var{n} @times{} GMP_NUMB_BITS}, but a smaller value might improve
performance if @var{M} or @var{A} are known to have leading zero bits.

This function requires scratch space of @code{mpn_sec_invert_itch(@var{n})}
limbs to be passed in the @var{tp} parameter.
@end deftypefun


@sp 1
@section Nails
@cindex Nails

@strong{Everything in this section is highly experimental and may disappear or
be subject to incompatible changes in a future version of GMP.}

Nails are an experimental feature whereby a few bits are left unused at the
top of each @code{mp_limb_t}.  This can significantly improve carry handling
on some processors.

All the @code{mpn} functions accepting limb data will expect the nail bits to
be zero on entry, and will return data with the nails similarly all zero.
This applies both to limb vectors and to single limb arguments.

Nails can be enabled by configuring with @samp{--enable-nails}.  By default
the number of bits will be chosen according to what suits the host processor,
but a particular number can be selected with @samp{--enable-nails=N}.

At the mpn level, a nail build is neither source nor binary compatible with a
non-nail build, strictly speaking.  But programs acting on limbs only through
the mpn functions are likely to work equally well with either build, and
judicious use of the definitions below should make any program compatible with
either build, at the source level.

For the higher level routines, meaning @code{mpz} etc, a nail build should be
fully source and binary compatible with a non-nail build.

@defmac GMP_NAIL_BITS
@defmacx GMP_NUMB_BITS
@defmacx GMP_LIMB_BITS
@code{GMP_NAIL_BITS} is the number of nail bits, or 0 when nails are not in
use.  @code{GMP_NUMB_BITS} is the number of data bits in a limb.
@code{GMP_LIMB_BITS} is the total number of bits in an @code{mp_limb_t}.  In
all cases

@example
GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
@end example
@end defmac

@defmac GMP_NAIL_MASK
@defmacx GMP_NUMB_MASK
Bit masks for the nail and number parts of a limb.  @code{GMP_NAIL_MASK} is 0
when nails are not in use.

@code{GMP_NAIL_MASK} is not often needed, since the nail part can be obtained
with @code{x >> GMP_NUMB_BITS}, and that means one less large constant, which
can help various RISC chips.
@end defmac

@defmac GMP_NUMB_MAX
The maximum value that can be stored in the number part of a limb.  This is
the same as @code{GMP_NUMB_MASK}, but can be used for clarity when doing
comparisons rather than bit-wise operations.
@end defmac

The term ``nails'' comes from finger or toe nails, which are at the ends of a
limb (arm or leg).  ``numb'' is short for number, but is also how the
developers felt after trying for a long time to come up with sensible names
for these things.

In the future (the distant future most likely) a non-zero nail might be
permitted, giving non-unique representations for numbers in a limb vector.
This would help vector processors since carries would only ever need to
propagate one or two limbs.


@node Random Number Functions, Formatted Output, Low-level Functions, Top
@chapter Random Number Functions
@cindex Random number functions

Sequences of pseudo-random numbers in GMP are generated using a variable of
type @code{gmp_randstate_t}, which holds an algorithm selection and a current
state.  Such a variable must be initialized by a call to one of the
@code{gmp_randinit} functions, and can be seeded with one of the
@code{gmp_randseed} functions.

The functions actually generating random numbers are described in @ref{Integer
Random Numbers}, and @ref{Miscellaneous Float Functions}.

The older style random number functions don't accept a @code{gmp_randstate_t}
parameter but instead share a global variable of that type.  They use a
default algorithm and are currently not seeded (though perhaps that will
change in the future).  The new functions accepting a @code{gmp_randstate_t}
are recommended for applications that care about randomness.

@menu
* Random State Initialization::
* Random State Seeding::
* Random State Miscellaneous::
@end menu

@node Random State Initialization, Random State Seeding, Random Number Functions, Random Number Functions
@section Random State Initialization
@cindex Random number state
@cindex Initialization functions

@deftypefun void gmp_randinit_default (gmp_randstate_t @var{state})
Initialize @var{state} with a default algorithm.  This will be a compromise
between speed and randomness, and is recommended for applications with no
special requirements.  Currently this is @code{gmp_randinit_mt}.
@end deftypefun

@deftypefun void gmp_randinit_mt (gmp_randstate_t @var{state})
@cindex Mersenne twister random numbers
Initialize @var{state} for a Mersenne Twister algorithm.  This algorithm is
fast and has good randomness properties.
@end deftypefun

@deftypefun void gmp_randinit_lc_2exp (gmp_randstate_t @var{state}, const mpz_t @var{a}, @w{unsigned long @var{c}}, @w{mp_bitcnt_t @var{m2exp}})
@cindex Linear congruential random numbers
Initialize @var{state} with a linear congruential algorithm @m{X = (@var{a}X +
@var{c}) @bmod 2^{m2exp}, X = (@var{a}*X + @var{c}) mod 2^@var{m2exp}}.

The low bits of @math{X} in this algorithm are not very random.  The least
significant bit will have a period no more than 2, and the second bit no more
than 4, etc.  For this reason only the high half of each @math{X} is actually
used.

When a random number of more than @math{@var{m2exp}/2} bits is to be
generated, multiple iterations of the recurrence are used and the results
concatenated.
@end deftypefun

@deftypefun int gmp_randinit_lc_2exp_size (gmp_randstate_t @var{state}, mp_bitcnt_t @var{size})
@cindex Linear congruential random numbers
Initialize @var{state} for a linear congruential algorithm as per
@code{gmp_randinit_lc_2exp}.  @var{a}, @var{c} and @var{m2exp} are selected
from a table, chosen so that @var{size} bits (or more) of each @math{X} will
be used, i.e.@: @math{@var{m2exp}/2 @ge{} @var{size}}.

If successful the return value is non-zero.  If @var{size} is bigger than the
table data provides then the return value is zero.  The maximum @var{size}
currently supported is 128.
@end deftypefun

@deftypefun void gmp_randinit_set (gmp_randstate_t @var{rop}, gmp_randstate_t @var{op})
Initialize @var{rop} with a copy of the algorithm and state from @var{op}.
@end deftypefun

@c  Although gmp_randinit, gmp_errno and related constants are obsolete, we
@c  still put @findex entries for them, since they're still documented and
@c  someone might be looking them up when perusing old application code.

@deftypefun void gmp_randinit (gmp_randstate_t @var{state}, @w{gmp_randalg_t @var{alg}}, @dots{})
@strong{This function is obsolete.}

@findex GMP_RAND_ALG_LC
@findex GMP_RAND_ALG_DEFAULT
Initialize @var{state} with an algorithm selected by @var{alg}.  The only
choice is @code{GMP_RAND_ALG_LC}, which is @code{gmp_randinit_lc_2exp_size}
described above.  A third parameter of type @code{unsigned long} is required,
this is the @var{size} for that function.  @code{GMP_RAND_ALG_DEFAULT} or 0
are the same as @code{GMP_RAND_ALG_LC}.

@c  For reference, this is the only place gmp_errno has been documented, and
@c  due to being non thread safe we won't be adding to it's uses.
@findex gmp_errno
@findex GMP_ERROR_UNSUPPORTED_ARGUMENT
@findex GMP_ERROR_INVALID_ARGUMENT
@code{gmp_randinit} sets bits in the global variable @code{gmp_errno} to
indicate an error.  @code{GMP_ERROR_UNSUPPORTED_ARGUMENT} if @var{alg} is
unsupported, or @code{GMP_ERROR_INVALID_ARGUMENT} if the @var{size} parameter
is too big.  It may be noted this error reporting is not thread safe (a good
reason to use @code{gmp_randinit_lc_2exp_size} instead).
@end deftypefun

@deftypefun void gmp_randclear (gmp_randstate_t @var{state})
Free all memory occupied by @var{state}.
@end deftypefun


@node Random State Seeding, Random State Miscellaneous, Random State Initialization, Random Number Functions
@section Random State Seeding
@cindex Random number seeding
@cindex Seeding random numbers

@deftypefun void gmp_randseed (gmp_randstate_t @var{state}, const mpz_t @var{seed})
@deftypefunx void gmp_randseed_ui (gmp_randstate_t @var{state}, @w{unsigned long int @var{seed}})
Set an initial seed value into @var{state}.

The size of a seed determines how many different sequences of random numbers
that it's possible to generate.  The ``quality'' of the seed is the randomness
of a given seed compared to the previous seed used, and this affects the
randomness of separate number sequences.  The method for choosing a seed is
critical if the generated numbers are to be used for important applications,
such as generating cryptographic keys.

Traditionally the system time has been used to seed, but care needs to be
taken with this.  If an application seeds often and the resolution of the
system clock is low, then the same sequence of numbers might be repeated.
Also, the system time is quite easy to guess, so if unpredictability is
required then it should definitely not be the only source for the seed value.
On some systems there's a special device @file{/dev/random} which provides
random data better suited for use as a seed.
@end deftypefun


@node Random State Miscellaneous,  , Random State Seeding, Random Number Functions
@section Random State Miscellaneous

@deftypefun {unsigned long} gmp_urandomb_ui (gmp_randstate_t @var{state}, unsigned long @var{n})
Return a uniformly distributed random number of @var{n} bits, i.e.@: in the
range 0 to @m{2^n-1,2^@var{n}-1} inclusive.  @var{n} must be less than or
equal to the number of bits in an @code{unsigned long}.
@end deftypefun

@deftypefun {unsigned long} gmp_urandomm_ui (gmp_randstate_t @var{state}, unsigned long @var{n})
Return a uniformly distributed random number in the range 0 to
@math{@var{n}-1}, inclusive.
@end deftypefun


@node Formatted Output, Formatted Input, Random Number Functions, Top
@chapter Formatted Output
@cindex Formatted output
@cindex @code{printf} formatted output

@menu
* Formatted Output Strings::
* Formatted Output Functions::
* C++ Formatted Output::
@end menu

@node Formatted Output Strings, Formatted Output Functions, Formatted Output, Formatted Output
@section Format Strings

@code{gmp_printf} and friends accept format strings similar to the standard C
@code{printf} (@pxref{Formatted Output,, Formatted Output, libc, The GNU C
Library Reference Manual}).  A format specification is of the form

@example
% [flags] [width] [.[precision]] [type] conv
@end example

GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
and @code{mpf_t} respectively, @samp{M} for @code{mp_limb_t}, and @samp{N} for
an @code{mp_limb_t} array.  @samp{Z}, @samp{Q}, @samp{M} and @samp{N} behave
like integers.  @samp{Q} will print a @samp{/} and a denominator, if needed.
@samp{F} behaves like a float.  For example,

@example
mpz_t z;
gmp_printf ("%s is an mpz %Zd\n", "here", z);

mpq_t q;
gmp_printf ("a hex rational: %#40Qx\n", q);

mpf_t f;
int   n;
gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);

mp_limb_t l;
gmp_printf ("limb %Mu\n", l);

const mp_limb_t *ptr;
mp_size_t       size;
gmp_printf ("limb array %Nx\n", ptr, size);
@end example

For @samp{N} the limbs are expected least significant first, as per the
@code{mpn} functions (@pxref{Low-level Functions}).  A negative size can be
given to print the value as a negative.

All the standard C @code{printf} types behave the same as the C library
@code{printf}, and can be freely intermixed with the GMP extensions.  In the
current implementation the standard parts of the format string are simply
handed to @code{printf} and only the GMP extensions handled directly.

The flags accepted are as follows.  GLIBC style @nisamp{'} is only for the
standard C types (not the GMP types), and only if the C library supports it.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{0} @tab pad with zeros (rather than spaces)
@item @nicode{#} @tab show the base with @samp{0x}, @samp{0X} or @samp{0}
@item @nicode{+} @tab always show a sign
@item (space)    @tab show a space or a @samp{-} sign
@item @nicode{'} @tab group digits, GLIBC style (not GMP types)
@end multitable
@end quotation

The optional width and precision can be given as a number within the format
string, or as a @samp{*} to take an extra parameter of type @code{int}, the
same as the standard @code{printf}.

The standard types accepted are as follows.  @samp{h} and @samp{l} are
portable, the rest will depend on the compiler (or include files) for the type
and the C library for the output.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{h}  @tab @nicode{short}
@item @nicode{hh} @tab @nicode{char}
@item @nicode{j}  @tab @nicode{intmax_t} or @nicode{uintmax_t}
@item @nicode{l}  @tab @nicode{long} or @nicode{wchar_t}
@item @nicode{ll} @tab @nicode{long long}
@item @nicode{L}  @tab @nicode{long double}
@item @nicode{q}  @tab @nicode{quad_t} or @nicode{u_quad_t}
@item @nicode{t}  @tab @nicode{ptrdiff_t}
@item @nicode{z}  @tab @nicode{size_t}
@end multitable
@end quotation

@noindent
The GMP types are

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{F}  @tab @nicode{mpf_t}, float conversions
@item @nicode{Q}  @tab @nicode{mpq_t}, integer conversions
@item @nicode{M}  @tab @nicode{mp_limb_t}, integer conversions
@item @nicode{N}  @tab @nicode{mp_limb_t} array, integer conversions
@item @nicode{Z}  @tab @nicode{mpz_t}, integer conversions
@end multitable
@end quotation

The conversions accepted are as follows.  @samp{a} and @samp{A} are always
supported for @code{mpf_t} but depend on the C library for standard C float
types.  @samp{m} and @samp{p} depend on the C library.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{a} @nicode{A} @tab hex floats, C99 style
@item @nicode{c}            @tab character
@item @nicode{d}            @tab decimal integer
@item @nicode{e} @nicode{E} @tab scientific format float
@item @nicode{f}            @tab fixed point float
@item @nicode{i}            @tab same as @nicode{d}
@item @nicode{g} @nicode{G} @tab fixed or scientific float
@item @nicode{m}            @tab @code{strerror} string, GLIBC style
@item @nicode{n}            @tab store characters written so far
@item @nicode{o}            @tab octal integer
@item @nicode{p}            @tab pointer
@item @nicode{s}            @tab string
@item @nicode{u}            @tab unsigned integer
@item @nicode{x} @nicode{X} @tab hex integer
@end multitable
@end quotation

@samp{o}, @samp{x} and @samp{X} are unsigned for the standard C types, but for
types @samp{Z}, @samp{Q} and @samp{N} they are signed.  @samp{u} is not
meaningful for @samp{Z}, @samp{Q} and @samp{N}.

@samp{M} is a proxy for the C library @samp{l} or @samp{L}, according to the
size of @code{mp_limb_t}.  Unsigned conversions will be usual, but a signed
conversion can be used and will interpret the value as a twos complement
negative.

@samp{n} can be used with any type, even the GMP types.

Other types or conversions that might be accepted by the C library
@code{printf} cannot be used through @code{gmp_printf}, this includes for
instance extensions registered with GLIBC @code{register_printf_function}.
Also currently there's no support for POSIX @samp{$} style numbered arguments
(perhaps this will be added in the future).

The precision field has its usual meaning for integer @samp{Z} and float
@samp{F} types, but is currently undefined for @samp{Q} and should not be used
with that.

@code{mpf_t} conversions only ever generate as many digits as can be
accurately represented by the operand, the same as @code{mpf_get_str} does.
Zeros will be used if necessary to pad to the requested precision.  This
happens even for an @samp{f} conversion of an @code{mpf_t} which is an
integer, for instance @math{2^@W{1024}} in an @code{mpf_t} of 128 bits
precision will only produce about 40 digits, then pad with zeros to the
decimal point.  An empty precision field like @samp{%.Fe} or @samp{%.Ff} can
be used to specifically request just the significant digits.  Without any dot
and thus no precision field, a precision value of 6 will be used.  Note that
these rules mean that @samp{%Ff}, @samp{%.Ff}, and @samp{%.0Ff} will all be
different.

The decimal point character (or string) is taken from the current locale
settings on systems which provide @code{localeconv} (@pxref{Locales,, Locales
and Internationalization, libc, The GNU C Library Reference Manual}).  The C
library will normally do the same for standard float output.

The format string is only interpreted as plain @code{char}s, multibyte
characters are not recognised.  Perhaps this will change in the future.


@node Formatted Output Functions, C++ Formatted Output, Formatted Output Strings, Formatted Output
@section Functions
@cindex Output functions

Each of the following functions is similar to the corresponding C library
function.  The basic @code{printf} forms take a variable argument list.  The
@code{vprintf} forms take an argument pointer, see @ref{Variadic Functions,,
Variadic Functions, libc, The GNU C Library Reference Manual}, or @samp{man 3
va_start}.

It should be emphasised that if a format string is invalid, or the arguments
don't match what the format specifies, then the behaviour of any of these
functions will be unpredictable.  GCC format string checking is not available,
since it doesn't recognise the GMP extensions.

The file based functions @code{gmp_printf} and @code{gmp_fprintf} will return
@math{-1} to indicate a write error.  Output is not ``atomic'', so partial
output may be produced if a write error occurs.  All the functions can return
@math{-1} if the C library @code{printf} variant in use returns @math{-1}, but
this shouldn't normally occur.

@deftypefun int gmp_printf (const char *@var{fmt}, @dots{})
@deftypefunx int gmp_vprintf (const char *@var{fmt}, va_list @var{ap})
Print to the standard output @code{stdout}.  Return the number of characters
written, or @math{-1} if an error occurred.
@end deftypefun

@deftypefun int gmp_fprintf (FILE *@var{fp}, const char *@var{fmt}, @dots{})
@deftypefunx int gmp_vfprintf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap})
Print to the stream @var{fp}.  Return the number of characters written, or
@math{-1} if an error occurred.
@end deftypefun

@deftypefun int gmp_sprintf (char *@var{buf}, const char *@var{fmt}, @dots{})
@deftypefunx int gmp_vsprintf (char *@var{buf}, const char *@var{fmt}, va_list @var{ap})
Form a null-terminated string in @var{buf}.  Return the number of characters
written, excluding the terminating null.

No overlap is permitted between the space at @var{buf} and the string
@var{fmt}.

These functions are not recommended, since there's no protection against
exceeding the space available at @var{buf}.
@end deftypefun

@deftypefun int gmp_snprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, @dots{})
@deftypefunx int gmp_vsnprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, va_list @var{ap})
Form a null-terminated string in @var{buf}.  No more than @var{size} bytes
will be written.  To get the full output, @var{size} must be enough for the
string and null-terminator.

The return value is the total number of characters which ought to have been
produced, excluding the terminating null.  If @math{@var{retval} @ge{}
@var{size}} then the actual output has been truncated to the first
@math{@var{size}-1} characters, and a null appended.

No overlap is permitted between the region @{@var{buf},@var{size}@} and the
@var{fmt} string.

Notice the return value is in ISO C99 @code{snprintf} style.  This is so even
if the C library @code{vsnprintf} is the older GLIBC 2.0.x style.
@end deftypefun

@deftypefun int gmp_asprintf (char **@var{pp}, const char *@var{fmt}, @dots{})
@deftypefunx int gmp_vasprintf (char **@var{pp}, const char *@var{fmt}, va_list @var{ap})
Form a null-terminated string in a block of memory obtained from the current
memory allocation function (@pxref{Custom Allocation}).  The block will be the
size of the string and null-terminator.  The address of the block in stored to
*@var{pp}.  The return value is the number of characters produced, excluding
the null-terminator.

Unlike the C library @code{asprintf}, @code{gmp_asprintf} doesn't return
@math{-1} if there's no more memory available, it lets the current allocation
function handle that.
@end deftypefun

@deftypefun int gmp_obstack_printf (struct obstack *@var{ob}, const char *@var{fmt}, @dots{})
@deftypefunx int gmp_obstack_vprintf (struct obstack *@var{ob}, const char *@var{fmt}, va_list @var{ap})
@cindex @code{obstack} output
Append to the current object in @var{ob}.  The return value is the number of
characters written.  A null-terminator is not written.

@var{fmt} cannot be within the current object in @var{ob}, since that object
might move as it grows.

These functions are available only when the C library provides the obstack
feature, which probably means only on GNU systems, see @ref{Obstacks,,
Obstacks, libc, The GNU C Library Reference Manual}.
@end deftypefun


@node C++ Formatted Output,  , Formatted Output Functions, Formatted Output
@section C++ Formatted Output
@cindex C++ @code{ostream} output
@cindex @code{ostream} output

The following functions are provided in @file{libgmpxx} (@pxref{Headers and
Libraries}), which is built if C++ support is enabled (@pxref{Build Options}).
Prototypes are available from @code{<gmp.h>}.

@deftypefun ostream& operator<< (ostream& @var{stream}, const mpz_t @var{op})
Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
@code{ios::width} is reset to 0 after output, the same as the standard
@code{ostream operator<<} routines do.

In hex or octal, @var{op} is printed as a signed number, the same as for
decimal.  This is unlike the standard @code{operator<<} routines on @code{int}
etc, which instead give twos complement.
@end deftypefun

@deftypefun ostream& operator<< (ostream& @var{stream}, const mpq_t @var{op})
Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
@code{ios::width} is reset to 0 after output, the same as the standard
@code{ostream operator<<} routines do.

Output will be a fraction like @samp{5/9}, or if the denominator is 1 then
just a plain integer like @samp{123}.

In hex or octal, @var{op} is printed as a signed value, the same as for
decimal.  If @code{ios::showbase} is set then a base indicator is shown on
both the numerator and denominator (if the denominator is required).
@end deftypefun

@deftypefun ostream& operator<< (ostream& @var{stream}, const mpf_t @var{op})
Print @var{op} to @var{stream}, using its @code{ios} formatting settings.
@code{ios::width} is reset to 0 after output, the same as the standard
@code{ostream operator<<} routines do.

The decimal point follows the standard library float @code{operator<<}, which
on recent systems means the @code{std::locale} imbued on @var{stream}.

Hex and octal are supported, unlike the standard @code{operator<<} on
@code{double}.  The mantissa will be in hex or octal, the exponent will be in
decimal.  For hex the exponent delimiter is an @samp{@@}.  This is as per
@code{mpf_out_str}.

@code{ios::showbase} is supported, and will put a base on the mantissa, for
example hex @samp{0x1.8} or @samp{0x0.8}, or octal @samp{01.4} or @samp{00.4}.
This last form is slightly strange, but at least differentiates itself from
decimal.
@end deftypefun

These operators mean that GMP types can be printed in the usual C++ way, for
example,

@example
mpz_t  z;
int    n;
...
cout << "iteration " << n << " value " << z << "\n";
@end example

But note that @code{ostream} output (and @code{istream} input, @pxref{C++
Formatted Input}) is the only overloading available for the GMP types and that
for instance using @code{+} with an @code{mpz_t} will have unpredictable
results.  For classes with overloading, see @ref{C++ Class Interface}.


@node Formatted Input, C++ Class Interface, Formatted Output, Top
@chapter Formatted Input
@cindex Formatted input
@cindex @code{scanf} formatted input

@menu
* Formatted Input Strings::
* Formatted Input Functions::
* C++ Formatted Input::
@end menu


@node Formatted Input Strings, Formatted Input Functions, Formatted Input, Formatted Input
@section Formatted Input Strings

@code{gmp_scanf} and friends accept format strings similar to the standard C
@code{scanf} (@pxref{Formatted Input,, Formatted Input, libc, The GNU C
Library Reference Manual}).  A format specification is of the form

@example
% [flags] [width] [type] conv
@end example

GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
and @code{mpf_t} respectively.  @samp{Z} and @samp{Q} behave like integers.
@samp{Q} will read a @samp{/} and a denominator, if present.  @samp{F} behaves
like a float.

GMP variables don't require an @code{&} when passed to @code{gmp_scanf}, since
they're already ``call-by-reference''.  For example,

@example
/* to read say "a(5) = 1234" */
int   n;
mpz_t z;
gmp_scanf ("a(%d) = %Zd\n", &n, z);

mpq_t q1, q2;
gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);

/* to read say "topleft (1.55,-2.66)" */
mpf_t x, y;
char  buf[32];
gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
@end example

All the standard C @code{scanf} types behave the same as in the C library
@code{scanf}, and can be freely intermixed with the GMP extensions.  In the
current implementation the standard parts of the format string are simply
handed to @code{scanf} and only the GMP extensions handled directly.

The flags accepted are as follows.  @samp{a} and @samp{'} will depend on
support from the C library, and @samp{'} cannot be used with GMP types.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{*} @tab read but don't store
@item @nicode{a} @tab allocate a buffer (string conversions)
@item @nicode{'} @tab grouped digits, GLIBC style (not GMP types)
@end multitable
@end quotation

The standard types accepted are as follows.  @samp{h} and @samp{l} are
portable, the rest will depend on the compiler (or include files) for the type
and the C library for the input.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{h}  @tab @nicode{short}
@item @nicode{hh} @tab @nicode{char}
@item @nicode{j}  @tab @nicode{intmax_t} or @nicode{uintmax_t}
@item @nicode{l}  @tab @nicode{long int}, @nicode{double} or @nicode{wchar_t}
@item @nicode{ll} @tab @nicode{long long}
@item @nicode{L}  @tab @nicode{long double}
@item @nicode{q}  @tab @nicode{quad_t} or @nicode{u_quad_t}
@item @nicode{t}  @tab @nicode{ptrdiff_t}
@item @nicode{z}  @tab @nicode{size_t}
@end multitable
@end quotation

@noindent
The GMP types are

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{F}  @tab @nicode{mpf_t}, float conversions
@item @nicode{Q}  @tab @nicode{mpq_t}, integer conversions
@item @nicode{Z}  @tab @nicode{mpz_t}, integer conversions
@end multitable
@end quotation

The conversions accepted are as follows.  @samp{p} and @samp{[} will depend on
support from the C library, the rest are standard.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{c}            @tab character or characters
@item @nicode{d}            @tab decimal integer
@item @nicode{e} @nicode{E} @nicode{f} @nicode{g} @nicode{G}
                            @tab float
@item @nicode{i}            @tab integer with base indicator
@item @nicode{n}            @tab characters read so far
@item @nicode{o}            @tab octal integer
@item @nicode{p}            @tab pointer
@item @nicode{s}            @tab string of non-whitespace characters
@item @nicode{u}            @tab decimal integer
@item @nicode{x} @nicode{X} @tab hex integer
@item @nicode{[}            @tab string of characters in a set
@end multitable
@end quotation

@samp{e}, @samp{E}, @samp{f}, @samp{g} and @samp{G} are identical, they all
read either fixed point or scientific format, and either upper or lower case
@samp{e} for the exponent in scientific format.

C99 style hex float format (@code{printf %a}, @pxref{Formatted Output
Strings}) is always accepted for @code{mpf_t}, but for the standard float
types it will depend on the C library.

@samp{x} and @samp{X} are identical, both accept both upper and lower case
hexadecimal.

@samp{o}, @samp{u}, @samp{x} and @samp{X} all read positive or negative
values.  For the standard C types these are described as ``unsigned''
conversions, but that merely affects certain overflow handling, negatives are
still allowed (per @code{strtoul}, @pxref{Parsing of Integers,, Parsing of
Integers, libc, The GNU C Library Reference Manual}).  For GMP types there are
no overflows, so @samp{d} and @samp{u} are identical.

@samp{Q} type reads the numerator and (optional) denominator as given.  If the
value might not be in canonical form then @code{mpq_canonicalize} must be
called before using it in any calculations (@pxref{Rational Number
Functions}).

@samp{Qi} will read a base specification separately for the numerator and
denominator.  For example @samp{0x10/11} would be 16/11, whereas
@samp{0x10/0x11} would be 16/17.

@samp{n} can be used with any of the types above, even the GMP types.
@samp{*} to suppress assignment is allowed, though in that case it would do
nothing at all.

Other conversions or types that might be accepted by the C library
@code{scanf} cannot be used through @code{gmp_scanf}.

Whitespace is read and discarded before a field, except for @samp{c} and
@samp{[} conversions.

For float conversions, the decimal point character (or string) expected is
taken from the current locale settings on systems which provide
@code{localeconv} (@pxref{Locales,, Locales and Internationalization, libc,
The GNU C Library Reference Manual}).  The C library will normally do the same
for standard float input.

The format string is only interpreted as plain @code{char}s, multibyte
characters are not recognised.  Perhaps this will change in the future.


@node Formatted Input Functions, C++ Formatted Input, Formatted Input Strings, Formatted Input
@section Formatted Input Functions
@cindex Input functions

Each of the following functions is similar to the corresponding C library
function.  The plain @code{scanf} forms take a variable argument list.  The
@code{vscanf} forms take an argument pointer, see @ref{Variadic Functions,,
Variadic Functions, libc, The GNU C Library Reference Manual}, or @samp{man 3
va_start}.

It should be emphasised that if a format string is invalid, or the arguments
don't match what the format specifies, then the behaviour of any of these
functions will be unpredictable.  GCC format string checking is not available,
since it doesn't recognise the GMP extensions.

No overlap is permitted between the @var{fmt} string and any of the results
produced.

@deftypefun int gmp_scanf (const char *@var{fmt}, @dots{})
@deftypefunx int gmp_vscanf (const char *@var{fmt}, va_list @var{ap})
Read from the standard input @code{stdin}.
@end deftypefun

@deftypefun int gmp_fscanf (FILE *@var{fp}, const char *@var{fmt}, @dots{})
@deftypefunx int gmp_vfscanf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap})
Read from the stream @var{fp}.
@end deftypefun

@deftypefun int gmp_sscanf (const char *@var{s}, const char *@var{fmt}, @dots{})
@deftypefunx int gmp_vsscanf (const char *@var{s}, const char *@var{fmt}, va_list @var{ap})
Read from a null-terminated string @var{s}.
@end deftypefun

The return value from each of these functions is the same as the standard C99
@code{scanf}, namely the number of fields successfully parsed and stored.
@samp{%n} fields and fields read but suppressed by @samp{*} don't count
towards the return value.

If end of input (or a file error) is reached before a character for a field or
a literal, and if no previous non-suppressed fields have matched, then the
return value is @code{EOF} instead of 0.  A whitespace character in the format
string is only an optional match and doesn't induce an @code{EOF} in this
fashion.  Leading whitespace read and discarded for a field don't count as
characters for that field.

For the GMP types, input parsing follows C99 rules, namely one character of
lookahead is used and characters are read while they continue to meet the
format requirements.  If this doesn't provide a complete number then the
function terminates, with that field not stored nor counted towards the return
value.  For instance with @code{mpf_t} an input @samp{1.23e-XYZ} would be read
up to the @samp{X} and that character pushed back since it's not a digit.  The
string @samp{1.23e-} would then be considered invalid since an @samp{e} must
be followed by at least one digit.

For the standard C types, in the current implementation GMP calls the C
library @code{scanf} functions, which might have looser rules about what
constitutes a valid input.

Note that @code{gmp_sscanf} is the same as @code{gmp_fscanf} and only does one
character of lookahead when parsing.  Although clearly it could look at its
entire input, it is deliberately made identical to @code{gmp_fscanf}, the same
way C99 @code{sscanf} is the same as @code{fscanf}.


@node C++ Formatted Input,  , Formatted Input Functions, Formatted Input
@section C++ Formatted Input
@cindex C++ @code{istream} input
@cindex @code{istream} input

The following functions are provided in @file{libgmpxx} (@pxref{Headers and
Libraries}), which is built only if C++ support is enabled (@pxref{Build
Options}).  Prototypes are available from @code{<gmp.h>}.

@deftypefun istream& operator>> (istream& @var{stream}, mpz_t @var{rop})
Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.
@end deftypefun

@deftypefun istream& operator>> (istream& @var{stream}, mpq_t @var{rop})
An integer like @samp{123} will be read, or a fraction like @samp{5/9}.  No
whitespace is allowed around the @samp{/}.  If the fraction is not in
canonical form then @code{mpq_canonicalize} must be called (@pxref{Rational
Number Functions}) before operating on it.

As per integer input, an @samp{0} or @samp{0x} base indicator is read when
none of @code{ios::dec}, @code{ios::oct} or @code{ios::hex} are set.  This is
done separately for numerator and denominator, so that for instance
@samp{0x10/11} is @math{16/11} and @samp{0x10/0x11} is @math{16/17}.
@end deftypefun

@deftypefun istream& operator>> (istream& @var{stream}, mpf_t @var{rop})
Read @var{rop} from @var{stream}, using its @code{ios} formatting settings.

Hex or octal floats are not supported, but might be in the future, or perhaps
it's best to accept only what the standard float @code{operator>>} does.
@end deftypefun

Note that digit grouping specified by the @code{istream} locale is currently
not accepted.  Perhaps this will change in the future.

@sp 1
These operators mean that GMP types can be read in the usual C++ way, for
example,

@example
mpz_t  z;
...
cin >> z;
@end example

But note that @code{istream} input (and @code{ostream} output, @pxref{C++
Formatted Output}) is the only overloading available for the GMP types and
that for instance using @code{+} with an @code{mpz_t} will have unpredictable
results.  For classes with overloading, see @ref{C++ Class Interface}.



@node C++ Class Interface, Custom Allocation, Formatted Input, Top
@chapter C++ Class Interface
@cindex C++ interface

This chapter describes the C++ class based interface to GMP.

All GMP C language types and functions can be used in C++ programs, since
@file{gmp.h} has @code{extern "C"} qualifiers, but the class interface offers
overloaded functions and operators which may be more convenient.

Due to the implementation of this interface, a reasonably recent C++ compiler
is required, one supporting namespaces, partial specialization of templates
and member templates.

@strong{Everything described in this chapter is to be considered preliminary
and might be subject to incompatible changes if some unforeseen difficulty
reveals itself.}

@menu
* C++ Interface General::
* C++ Interface Integers::
* C++ Interface Rationals::
* C++ Interface Floats::
* C++ Interface Random Numbers::
* C++ Interface Limitations::
@end menu


@node C++ Interface General, C++ Interface Integers, C++ Class Interface, C++ Class Interface
@section C++ Interface General

@noindent
All the C++ classes and functions are available with

@cindex @code{gmpxx.h}
@example
#include <gmpxx.h>
@end example

Programs should be linked with the @file{libgmpxx} and @file{libgmp}
libraries.  For example,

@example
g++ mycxxprog.cc -lgmpxx -lgmp
@end example

@noindent
The classes defined are

@deftp Class mpz_class
@deftpx Class mpq_class
@deftpx Class mpf_class
@end deftp

The standard operators and various standard functions are overloaded to allow
arithmetic with these classes.  For example,

@example
int
main (void)
@{
  mpz_class a, b, c;

  a = 1234;
  b = "-5678";
  c = a+b;
  cout << "sum is " << c << "\n";
  cout << "absolute value is " << abs(c) << "\n";

  return 0;
@}
@end example

An important feature of the implementation is that an expression like
@code{a=b+c} results in a single call to the corresponding @code{mpz_add},
without using a temporary for the @code{b+c} part.  Expressions which by their
nature imply intermediate values, like @code{a=b*c+d*e}, still use temporaries
though.

The classes can be freely intermixed in expressions, as can the classes and
the standard types @code{long}, @code{unsigned long} and @code{double}.
Smaller types like @code{int} or @code{float} can also be intermixed, since
C++ will promote them.

Note that @code{bool} is not accepted directly, but must be explicitly cast to
an @code{int} first.  This is because C++ will automatically convert any
pointer to a @code{bool}, so if GMP accepted @code{bool} it would make all
sorts of invalid class and pointer combinations compile but almost certainly
not do anything sensible.

Conversions back from the classes to standard C++ types aren't done
automatically, instead member functions like @code{get_si} are provided (see
the following sections for details).

Also there are no automatic conversions from the classes to the corresponding
GMP C types, instead a reference to the underlying C object can be obtained
with the following functions,

@deftypefun mpz_t mpz_class::get_mpz_t ()
@deftypefunx mpq_t mpq_class::get_mpq_t ()
@deftypefunx mpf_t mpf_class::get_mpf_t ()
@end deftypefun

These can be used to call a C function which doesn't have a C++ class
interface.  For example to set @code{a} to the GCD of @code{b} and @code{c},

@example
mpz_class a, b, c;
...
mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
@end example

In the other direction, a class can be initialized from the corresponding GMP
C type, or assigned to if an explicit constructor is used.  In both cases this
makes a copy of the value, it doesn't create any sort of association.  For
example,

@example
mpz_t z;
// ... init and calculate z ...
mpz_class x(z);
mpz_class y;
y = mpz_class (z);
@end example

There are no namespace setups in @file{gmpxx.h}, all types and functions are
simply put into the global namespace.  This is what @file{gmp.h} has done in
the past, and continues to do for compatibility.  The extras provided by
@file{gmpxx.h} follow GMP naming conventions and are unlikely to clash with
anything.


@node C++ Interface Integers, C++ Interface Rationals, C++ Interface General, C++ Class Interface
@section C++ Interface Integers

@deftypefun {} mpz_class::mpz_class (type @var{n})
Construct an @code{mpz_class}.  All the standard C++ types may be used, except
@code{long long} and @code{long double}, and all the GMP C++ classes can be
used, although conversions from @code{mpq_class} and @code{mpf_class} are
@code{explicit}.  Any necessary conversion follows the corresponding C
function, for example @code{double} follows @code{mpz_set_d}
(@pxref{Assigning Integers}).
@end deftypefun

@deftypefun explicit mpz_class::mpz_class (const mpz_t @var{z})
Construct an @code{mpz_class} from an @code{mpz_t}.  The value in @var{z} is
copied into the new @code{mpz_class}, there won't be any permanent association
between it and @var{z}.
@end deftypefun

@deftypefun explicit mpz_class::mpz_class (const char *@var{s}, int @var{base} = 0)
@deftypefunx explicit mpz_class::mpz_class (const string& @var{s}, int @var{base} = 0)
Construct an @code{mpz_class} converted from a string using @code{mpz_set_str}
(@pxref{Assigning Integers}).

If the string is not a valid integer, an @code{std::invalid_argument}
exception is thrown.  The same applies to @code{operator=}.
@end deftypefun

@deftypefun mpz_class operator"" _mpz (const char *@var{str})
With C++11 compilers, integers can be constructed with the syntax
@code{123_mpz} which is equivalent to @code{mpz_class("123")}.
@end deftypefun

@deftypefun mpz_class operator/ (mpz_class @var{a}, mpz_class @var{d})
@deftypefunx mpz_class operator% (mpz_class @var{a}, mpz_class @var{d})
Divisions involving @code{mpz_class} round towards zero, as per the
@code{mpz_tdiv_q} and @code{mpz_tdiv_r} functions (@pxref{Integer Division}).
This is the same as the C99 @code{/} and @code{%} operators.

The @code{mpz_fdiv@dots{}} or @code{mpz_cdiv@dots{}} functions can always be called
directly if desired.  For example,

@example
mpz_class q, a, d;
...
mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
@end example
@end deftypefun

@deftypefun mpz_class abs (mpz_class @var{op})
@deftypefunx int cmp (mpz_class @var{op1}, type @var{op2})
@deftypefunx int cmp (type @var{op1}, mpz_class @var{op2})
@maybepagebreak
@deftypefunx bool mpz_class::fits_sint_p (void)
@deftypefunx bool mpz_class::fits_slong_p (void)
@deftypefunx bool mpz_class::fits_sshort_p (void)
@maybepagebreak
@deftypefunx bool mpz_class::fits_uint_p (void)
@deftypefunx bool mpz_class::fits_ulong_p (void)
@deftypefunx bool mpz_class::fits_ushort_p (void)
@maybepagebreak
@deftypefunx double mpz_class::get_d (void)
@deftypefunx long mpz_class::get_si (void)
@deftypefunx string mpz_class::get_str (int @var{base} = 10)
@deftypefunx {unsigned long} mpz_class::get_ui (void)
@maybepagebreak
@deftypefunx int mpz_class::set_str (const char *@var{str}, int @var{base})
@deftypefunx int mpz_class::set_str (const string& @var{str}, int @var{base})
@deftypefunx int sgn (mpz_class @var{op})
@deftypefunx mpz_class sqrt (mpz_class @var{op})
@maybepagebreak
@deftypefunx mpz_class gcd (mpz_class @var{op1}, mpz_class @var{op2})
@deftypefunx mpz_class lcm (mpz_class @var{op1}, mpz_class @var{op2})
@deftypefunx mpz_class mpz_class::factorial (type @var{op})
@deftypefunx mpz_class factorial (mpz_class @var{op})
@deftypefunx mpz_class mpz_class::primorial (type @var{op})
@deftypefunx mpz_class primorial (mpz_class @var{op})
@deftypefunx mpz_class mpz_class::fibonacci (type @var{op})
@deftypefunx mpz_class fibonacci (mpz_class @var{op})
@maybepagebreak
@deftypefunx void mpz_class::swap (mpz_class& @var{op})
@deftypefunx void swap (mpz_class& @var{op1}, mpz_class& @var{op2})
These functions provide a C++ class interface to the corresponding GMP C
routines.  Calling @code{factorial} or @code{primorial} on a negative number
is undefined.

@code{cmp} can be used with any of the classes or the standard C++ types,
except @code{long long} and @code{long double}.
@end deftypefun

@sp 1
Overloaded operators for combinations of @code{mpz_class} and @code{double}
are provided for completeness, but it should be noted that if the given
@code{double} is not an integer then the way any rounding is done is currently
unspecified.  The rounding might take place at the start, in the middle, or at
the end of the operation, and it might change in the future.

Conversions between @code{mpz_class} and @code{double}, however, are defined
to follow the corresponding C functions @code{mpz_get_d} and @code{mpz_set_d}.
And comparisons are always made exactly, as per @code{mpz_cmp_d}.


@node C++ Interface Rationals, C++ Interface Floats, C++ Interface Integers, C++ Class Interface
@section C++ Interface Rationals

In all the following constructors, if a fraction is given then it should be in
canonical form, or if not then @code{mpq_class::canonicalize} called.

@deftypefun {} mpq_class::mpq_class (type @var{op})
@deftypefunx {} mpq_class::mpq_class (integer @var{num}, integer @var{den})
Construct an @code{mpq_class}.  The initial value can be a single value of any
type (conversion from @code{mpf_class} is @code{explicit}), or a pair of
integers (@code{mpz_class} or standard C++ integer types) representing a
fraction, except that @code{long long} and @code{long double} are not
supported.  For example,

@example
mpq_class q (99);
mpq_class q (1.75);
mpq_class q (1, 3);
@end example
@end deftypefun

@deftypefun explicit mpq_class::mpq_class (const mpq_t @var{q})
Construct an @code{mpq_class} from an @code{mpq_t}.  The value in @var{q} is
copied into the new @code{mpq_class}, there won't be any permanent association
between it and @var{q}.
@end deftypefun

@deftypefun explicit mpq_class::mpq_class (const char *@var{s}, int @var{base} = 0)
@deftypefunx explicit mpq_class::mpq_class (const string& @var{s}, int @var{base} = 0)
Construct an @code{mpq_class} converted from a string using @code{mpq_set_str}
(@pxref{Initializing Rationals}).

If the string is not a valid rational, an @code{std::invalid_argument}
exception is thrown.  The same applies to @code{operator=}.
@end deftypefun

@deftypefun mpq_class operator"" _mpq (const char *@var{str})
With C++11 compilers, integral rationals can be constructed with the syntax
@code{123_mpq} which is equivalent to @code{mpq_class(123_mpz)}. Other
rationals can be built as @code{-1_mpq/2} or @code{0xb_mpq/123456_mpz}.
@end deftypefun

@deftypefun void mpq_class::canonicalize ()
Put an @code{mpq_class} into canonical form, as per @ref{Rational Number
Functions}.  All arithmetic operators require their operands in canonical
form, and will return results in canonical form.
@end deftypefun

@deftypefun mpq_class abs (mpq_class @var{op})
@deftypefunx int cmp (mpq_class @var{op1}, type @var{op2})
@deftypefunx int cmp (type @var{op1}, mpq_class @var{op2})
@maybepagebreak
@deftypefunx double mpq_class::get_d (void)
@deftypefunx string mpq_class::get_str (int @var{base} = 10)
@maybepagebreak
@deftypefunx int mpq_class::set_str (const char *@var{str}, int @var{base})
@deftypefunx int mpq_class::set_str (const string& @var{str}, int @var{base})
@deftypefunx int sgn (mpq_class @var{op})
@maybepagebreak
@deftypefunx void mpq_class::swap (mpq_class& @var{op})
@deftypefunx void swap (mpq_class& @var{op1}, mpq_class& @var{op2})
These functions provide a C++ class interface to the corresponding GMP C
routines.

@code{cmp} can be used with any of the classes or the standard C++ types,
except @code{long long} and @code{long double}.
@end deftypefun

@deftypefun {mpz_class&} mpq_class::get_num ()
@deftypefunx {mpz_class&} mpq_class::get_den ()
Get a reference to an @code{mpz_class} which is the numerator or denominator
of an @code{mpq_class}.  This can be used both for read and write access.  If
the object returned is modified, it modifies the original @code{mpq_class}.

If direct manipulation might produce a non-canonical value, then
@code{mpq_class::canonicalize} must be called before further operations.
@end deftypefun

@deftypefun mpz_t mpq_class::get_num_mpz_t ()
@deftypefunx mpz_t mpq_class::get_den_mpz_t ()
Get a reference to the underlying @code{mpz_t} numerator or denominator of an
@code{mpq_class}.  This can be passed to C functions expecting an
@code{mpz_t}.  Any modifications made to the @code{mpz_t} will modify the
original @code{mpq_class}.

If direct manipulation might produce a non-canonical value, then
@code{mpq_class::canonicalize} must be called before further operations.
@end deftypefun

@deftypefun istream& operator>> (istream& @var{stream}, mpq_class& @var{rop});
Read @var{rop} from @var{stream}, using its @code{ios} formatting settings,
the same as @code{mpq_t operator>>} (@pxref{C++ Formatted Input}).

If the @var{rop} read might not be in canonical form then
@code{mpq_class::canonicalize} must be called.
@end deftypefun


@node C++ Interface Floats, C++ Interface Random Numbers, C++ Interface Rationals, C++ Class Interface
@section C++ Interface Floats

When an expression requires the use of temporary intermediate @code{mpf_class}
values, like @code{f=g*h+x*y}, those temporaries will have the same precision
as the destination @code{f}.  Explicit constructors can be used if this
doesn't suit.

@deftypefun {} mpf_class::mpf_class (type @var{op})
@deftypefunx {} mpf_class::mpf_class (type @var{op}, mp_bitcnt_t @var{prec})
Construct an @code{mpf_class}.  Any standard C++ type can be used, except
@code{long long} and @code{long double}, and any of the GMP C++ classes can be
used.

If @var{prec} is given, the initial precision is that value, in bits.  If
@var{prec} is not given, then the initial precision is determined by the type
of @var{op} given.  An @code{mpz_class}, @code{mpq_class}, or C++
builtin type will give the default @code{mpf} precision (@pxref{Initializing
Floats}).  An @code{mpf_class} or expression will give the precision of that
value.  The precision of a binary expression is the higher of the two
operands.

@example
mpf_class f(1.5);        // default precision
mpf_class f(1.5, 500);   // 500 bits (at least)
mpf_class f(x);          // precision of x
mpf_class f(abs(x));     // precision of x
mpf_class f(-g, 1000);   // 1000 bits (at least)
mpf_class f(x+y);        // greater of precisions of x and y
@end example
@end deftypefun

@deftypefun explicit mpf_class::mpf_class (const mpf_t @var{f})
@deftypefunx {} mpf_class::mpf_class (const mpf_t @var{f}, mp_bitcnt_t @var{prec})
Construct an @code{mpf_class} from an @code{mpf_t}.  The value in @var{f} is
copied into the new @code{mpf_class}, there won't be any permanent association
between it and @var{f}.

If @var{prec} is given, the initial precision is that value, in bits.  If
@var{prec} is not given, then the initial precision is that of @var{f}.
@end deftypefun

@deftypefun explicit mpf_class::mpf_class (const char *@var{s})
@deftypefunx {} mpf_class::mpf_class (const char *@var{s}, mp_bitcnt_t @var{prec}, int @var{base} = 0)
@deftypefunx explicit mpf_class::mpf_class (const string& @var{s})
@deftypefunx {} mpf_class::mpf_class (const string& @var{s}, mp_bitcnt_t @var{prec}, int @var{base} = 0)
Construct an @code{mpf_class} converted from a string using @code{mpf_set_str}
(@pxref{Assigning Floats}).  If @var{prec} is given, the initial precision is
that value, in bits.  If not, the default @code{mpf} precision
(@pxref{Initializing Floats}) is used.

If the string is not a valid float, an @code{std::invalid_argument} exception
is thrown.  The same applies to @code{operator=}.
@end deftypefun

@deftypefun mpf_class operator"" _mpf (const char *@var{str})
With C++11 compilers, floats can be constructed with the syntax
@code{1.23e-1_mpf} which is equivalent to @code{mpf_class("1.23e-1")}.
@end deftypefun

@deftypefun {mpf_class&} mpf_class::operator= (type @var{op})
Convert and store the given @var{op} value to an @code{mpf_class} object.  The
same types are accepted as for the constructors above.

Note that @code{operator=} only stores a new value, it doesn't copy or change
the precision of the destination, instead the value is truncated if necessary.
This is the same as @code{mpf_set} etc.  Note in particular this means for
@code{mpf_class} a copy constructor is not the same as a default constructor
plus assignment.

@example
mpf_class x (y);   // x created with precision of y

mpf_class x;       // x created with default precision
x = y;             // value truncated to that precision
@end example

Applications using templated code may need to be careful about the assumptions
the code makes in this area, when working with @code{mpf_class} values of
various different or non-default precisions.  For instance implementations of
the standard @code{complex} template have been seen in both styles above,
though of course @code{complex} is normally only actually specified for use
with the builtin float types.
@end deftypefun

@deftypefun mpf_class abs (mpf_class @var{op})
@deftypefunx mpf_class ceil (mpf_class @var{op})
@deftypefunx int cmp (mpf_class @var{op1}, type @var{op2})
@deftypefunx int cmp (type @var{op1}, mpf_class @var{op2})
@maybepagebreak
@deftypefunx bool mpf_class::fits_sint_p (void)
@deftypefunx bool mpf_class::fits_slong_p (void)
@deftypefunx bool mpf_class::fits_sshort_p (void)
@maybepagebreak
@deftypefunx bool mpf_class::fits_uint_p (void)
@deftypefunx bool mpf_class::fits_ulong_p (void)
@deftypefunx bool mpf_class::fits_ushort_p (void)
@maybepagebreak
@deftypefunx mpf_class floor (mpf_class @var{op})
@deftypefunx mpf_class hypot (mpf_class @var{op1}, mpf_class @var{op2})
@maybepagebreak
@deftypefunx double mpf_class::get_d (void)
@deftypefunx long mpf_class::get_si (void)
@deftypefunx string mpf_class::get_str (mp_exp_t& @var{exp}, int @var{base} = 10, size_t @var{digits} = 0)
@deftypefunx {unsigned long} mpf_class::get_ui (void)
@maybepagebreak
@deftypefunx int mpf_class::set_str (const char *@var{str}, int @var{base})
@deftypefunx int mpf_class::set_str (const string& @var{str}, int @var{base})
@deftypefunx int sgn (mpf_class @var{op})
@deftypefunx mpf_class sqrt (mpf_class @var{op})
@maybepagebreak
@deftypefunx void mpf_class::swap (mpf_class& @var{op})
@deftypefunx void swap (mpf_class& @var{op1}, mpf_class& @var{op2})
@deftypefunx mpf_class trunc (mpf_class @var{op})
These functions provide a C++ class interface to the corresponding GMP C
routines.

@code{cmp} can be used with any of the classes or the standard C++ types,
except @code{long long} and @code{long double}.

The accuracy provided by @code{hypot} is not currently guaranteed.
@end deftypefun

@deftypefun {mp_bitcnt_t} mpf_class::get_prec ()
@deftypefunx void mpf_class::set_prec (mp_bitcnt_t @var{prec})
@deftypefunx void mpf_class::set_prec_raw (mp_bitcnt_t @var{prec})
Get or set the current precision of an @code{mpf_class}.

The restrictions described for @code{mpf_set_prec_raw} (@pxref{Initializing
Floats}) apply to @code{mpf_class::set_prec_raw}.  Note in particular that the
@code{mpf_class} must be restored to it's allocated precision before being
destroyed.  This must be done by application code, there's no automatic
mechanism for it.
@end deftypefun


@node C++ Interface Random Numbers, C++ Interface Limitations, C++ Interface Floats, C++ Class Interface
@section C++ Interface Random Numbers

@deftp Class gmp_randclass
The C++ class interface to the GMP random number functions uses
@code{gmp_randclass} to hold an algorithm selection and current state, as per
@code{gmp_randstate_t}.
@end deftp

@deftypefun {} gmp_randclass::gmp_randclass (void (*@var{randinit}) (gmp_randstate_t, @dots{}), @dots{})
Construct a @code{gmp_randclass}, using a call to the given @var{randinit}
function (@pxref{Random State Initialization}).  The arguments expected are
the same as @var{randinit}, but with @code{mpz_class} instead of @code{mpz_t}.
For example,

@example
gmp_randclass r1 (gmp_randinit_default);
gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
gmp_randclass r4 (gmp_randinit_mt);
@end example

@code{gmp_randinit_lc_2exp_size} will fail if the size requested is too big,
an @code{std::length_error} exception is thrown in that case.
@end deftypefun

@deftypefun {} gmp_randclass::gmp_randclass (gmp_randalg_t @var{alg}, @dots{})
Construct a @code{gmp_randclass} using the same parameters as
@code{gmp_randinit} (@pxref{Random State Initialization}).  This function is
obsolete and the above @var{randinit} style should be preferred.
@end deftypefun

@deftypefun void gmp_randclass::seed (unsigned long int @var{s})
@deftypefunx void gmp_randclass::seed (mpz_class @var{s})
Seed a random number generator.  See @pxref{Random Number Functions}, for how
to choose a good seed.
@end deftypefun

@deftypefun mpz_class gmp_randclass::get_z_bits (mp_bitcnt_t @var{bits})
@deftypefunx mpz_class gmp_randclass::get_z_bits (mpz_class @var{bits})
Generate a random integer with a specified number of bits.
@end deftypefun

@deftypefun mpz_class gmp_randclass::get_z_range (mpz_class @var{n})
Generate a random integer in the range 0 to @math{@var{n}-1} inclusive.
@end deftypefun

@deftypefun mpf_class gmp_randclass::get_f ()
@deftypefunx mpf_class gmp_randclass::get_f (mp_bitcnt_t @var{prec})
Generate a random float @var{f} in the range @math{0 <= @var{f} < 1}.  @var{f}
will be to @var{prec} bits precision, or if @var{prec} is not given then to
the precision of the destination.  For example,

@example
gmp_randclass  r;
...
mpf_class  f (0, 512);   // 512 bits precision
f = r.get_f();           // random number, 512 bits
@end example
@end deftypefun



@node C++ Interface Limitations,  , C++ Interface Random Numbers, C++ Class Interface
@section C++ Interface Limitations

@table @asis
@item @code{mpq_class} and Templated Reading
A generic piece of template code probably won't know that @code{mpq_class}
requires a @code{canonicalize} call if inputs read with @code{operator>>}
might be non-canonical.  This can lead to incorrect results.

@code{operator>>} behaves as it does for reasons of efficiency.  A
canonicalize can be quite time consuming on large operands, and is best
avoided if it's not necessary.

But this potential difficulty reduces the usefulness of @code{mpq_class}.
Perhaps a mechanism to tell @code{operator>>} what to do will be adopted in
the future, maybe a preprocessor define, a global flag, or an @code{ios} flag
pressed into service.  Or maybe, at the risk of inconsistency, the
@code{mpq_class} @code{operator>>} could canonicalize and leave @code{mpq_t}
@code{operator>>} not doing so, for use on those occasions when that's
acceptable.  Send feedback or alternate ideas to @email{gmp-bugs@@gmplib.org}.

@item Subclassing
Subclassing the GMP C++ classes works, but is not currently recommended.

Expressions involving subclasses resolve correctly (or seem to), but in normal
C++ fashion the subclass doesn't inherit constructors and assignments.
There's many of those in the GMP classes, and a good way to reestablish them
in a subclass is not yet provided.

@item Templated Expressions
A subtle difficulty exists when using expressions together with
application-defined template functions.  Consider the following, with @code{T}
intended to be some numeric type,

@example
template <class T>
T fun (const T &, const T &);
@end example

@noindent
When used with, say, plain @code{mpz_class} variables, it works fine: @code{T}
is resolved as @code{mpz_class}.

@example
mpz_class f(1), g(2);
fun (f, g);    // Good
@end example

@noindent
But when one of the arguments is an expression, it doesn't work.

@example
mpz_class f(1), g(2), h(3);
fun (f, g+h);  // Bad
@end example

This is because @code{g+h} ends up being a certain expression template type
internal to @code{gmpxx.h}, which the C++ template resolution rules are unable
to automatically convert to @code{mpz_class}.  The workaround is simply to add
an explicit cast.

@example
mpz_class f(1), g(2), h(3);
fun (f, mpz_class(g+h));  // Good
@end example

Similarly, within @code{fun} it may be necessary to cast an expression to type
@code{T} when calling a templated @code{fun2}.

@example
template <class T>
void fun (T f, T g)
@{
  fun2 (f, f+g);     // Bad
@}

template <class T>
void fun (T f, T g)
@{
  fun2 (f, T(f+g));  // Good
@}
@end example

@item C++11
C++11 provides several new ways in which types can be inferred: @code{auto},
@code{decltype}, etc. While they can be very convenient, they don't mix well
with expression templates. In this example, the addition is performed twice,
as if we had defined @code{sum} as a macro.

@example
mpz_class z = 33;
auto sum = z + z;
mpz_class prod = sum * sum;
@end example

This other example may crash, though some compilers might make it look like
it is working, because the expression @code{z+z} goes out of scope before it
is evaluated.

@example
mpz_class z = 33;
auto sum = z + z + z;
mpz_class prod = sum * 2;
@end example

It is thus strongly recommended to avoid @code{auto} anywhere a GMP C++
expression may appear.
@end table


@node Custom Allocation, Language Bindings, C++ Class Interface, Top
@comment  node-name,  next,  previous,  up
@chapter Custom Allocation
@cindex Custom allocation
@cindex Memory allocation
@cindex Allocation of memory

By default GMP uses @code{malloc}, @code{realloc} and @code{free} for memory
allocation, and if they fail GMP prints a message to the standard error output
and terminates the program.

Alternate functions can be specified, to allocate memory in a different way or
to have a different error action on running out of memory.

@deftypefun void mp_set_memory_functions (@* void *(*@var{alloc_func_ptr}) (size_t), @* void *(*@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (*@var{free_func_ptr}) (void *, size_t))
Replace the current allocation functions from the arguments.  If an argument
is @code{NULL}, the corresponding default function is used.

These functions will be used for all memory allocation done by GMP, apart from
temporary space from @code{alloca} if that function is available and GMP is
configured to use it (@pxref{Build Options}).

@strong{Be sure to call @code{mp_set_memory_functions} only when there are no
active GMP objects allocated using the previous memory functions!  Usually
that means calling it before any other GMP function.}
@end deftypefun

The functions supplied should fit the following declarations:

@deftypevr Function {void *} allocate_function (size_t @var{alloc_size})
Return a pointer to newly allocated space with at least @var{alloc_size}
bytes.
@end deftypevr

@deftypevr Function {void *} reallocate_function (void *@var{ptr}, size_t @var{old_size}, size_t @var{new_size})
Resize a previously allocated block @var{ptr} of @var{old_size} bytes to be
@var{new_size} bytes.

The block may be moved if necessary or if desired, and in that case the
smaller of @var{old_size} and @var{new_size} bytes must be copied to the new
location.  The return value is a pointer to the resized block, that being the
new location if moved or just @var{ptr} if not.

@var{ptr} is never @code{NULL}, it's always a previously allocated block.
@var{new_size} may be bigger or smaller than @var{old_size}.
@end deftypevr

@deftypevr Function void free_function (void *@var{ptr}, size_t @var{size})
De-allocate the space pointed to by @var{ptr}.

@var{ptr} is never @code{NULL}, it's always a previously allocated block of
@var{size} bytes.
@end deftypevr

A @dfn{byte} here means the unit used by the @code{sizeof} operator.

The @var{reallocate_function} parameter @var{old_size} and the
@var{free_function} parameter @var{size} are passed for convenience, but of
course they can be ignored if not needed by an implementation.  The default
functions using @code{malloc} and friends for instance don't use them.

No error return is allowed from any of these functions, if they return then
they must have performed the specified operation.  In particular note that
@var{allocate_function} or @var{reallocate_function} mustn't return
@code{NULL}.

Getting a different fatal error action is a good use for custom allocation
functions, for example giving a graphical dialog rather than the default print
to @code{stderr}.  How much is possible when genuinely out of memory is
another question though.

There's currently no defined way for the allocation functions to recover from
an error such as out of memory, they must terminate program execution.  A
@code{longjmp} or throwing a C++ exception will have undefined results.  This
may change in the future.

GMP may use allocated blocks to hold pointers to other allocated blocks.  This
will limit the assumptions a conservative garbage collection scheme can make.

Since the default GMP allocation uses @code{malloc} and friends, those
functions will be linked in even if the first thing a program does is an
@code{mp_set_memory_functions}.  It's necessary to change the GMP sources if
this is a problem.

@sp 1
@deftypefun void mp_get_memory_functions (@* void *(**@var{alloc_func_ptr}) (size_t), @* void *(**@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (**@var{free_func_ptr}) (void *, size_t))
Get the current allocation functions, storing function pointers to the
locations given by the arguments.  If an argument is @code{NULL}, that
function pointer is not stored.

@need 1000
For example, to get just the current free function,

@example
void (*freefunc) (void *, size_t);

mp_get_memory_functions (NULL, NULL, &freefunc);
@end example
@end deftypefun

@node Language Bindings, Algorithms, Custom Allocation, Top
@chapter Language Bindings
@cindex Language bindings
@cindex Other languages

The following packages and projects offer access to GMP from languages other
than C, though perhaps with varying levels of functionality and efficiency.

@c  @spaceuref{U} is the same as @uref{U}, but with a couple of extra spaces
@c  in tex, just to separate the URL from the preceding text a bit.
@iftex
@macro spaceuref {U}
@ @ @uref{\U\}
@end macro
@end iftex
@ifnottex
@macro spaceuref {U}
@uref{\U\}
@end macro
@end ifnottex

@sp 1
@table @asis
@item C++
@itemize @bullet
@item
GMP C++ class interface, @pxref{C++ Class Interface} @* Straightforward
interface, expression templates to eliminate temporaries.
@item
ALP @spaceuref{https://www-sop.inria.fr/saga/logiciels/ALP/} @* Linear algebra and
polynomials using templates.
@item
CLN @spaceuref{https://www.ginac.de/CLN/} @* High level classes for arithmetic.
@item
Linbox @spaceuref{http://www.linalg.org/} @* Sparse vectors and matrices.
@item
NTL @spaceuref{http://www.shoup.net/ntl/} @* A C++ number theory library.
@end itemize

@c @item D
@c @itemize @bullet
@c @item
@c gmp-d @spaceuref{http://home.comcast.net/~benhinkle/gmp-d/}
@c @end itemize

@item Eiffel
@itemize @bullet
@item
Eiffelroom @spaceuref{http://www.eiffelroom.org/node/442}
@end itemize

@c @item Fortran
@c @itemize @bullet
@c @item
@c Omni F77 @spaceuref{http://phase.hpcc.jp/Omni/home.html} @* Arbitrary
@c precision floats.
@c @end itemize

@item Haskell
@itemize @bullet
@item
Glasgow Haskell Compiler @spaceuref{https://www.haskell.org/ghc/}
@end itemize

@item Java
@itemize @bullet
@item
Kaffe @spaceuref{https://github.com/kaffe/kaffe}
@end itemize

@item Lisp
@itemize @bullet
@item
GNU Common Lisp @spaceuref{https://www.gnu.org/software/gcl/gcl.html}
@item
Librep @spaceuref{http://librep.sourceforge.net/}
@item
@c  FIXME: When there's a stable release with gmp support, just refer to it
@c  rather than bothering to talk about betas.
XEmacs (21.5.18 beta and up) @spaceuref{https://www.xemacs.org} @* Optional
big integers, rationals and floats using GMP.
@end itemize

@item ML
@itemize @bullet
@item
MLton compiler @spaceuref{http://mlton.org/}
@end itemize

@item Objective Caml
@itemize @bullet
@item
MLGMP @spaceuref{https://opam.ocaml.org/packages/mlgmp/}
@item
Numerix @spaceuref{http://pauillac.inria.fr/~quercia/} @* Optionally using
GMP.
@end itemize

@item Oz
@itemize @bullet
@item
Mozart @spaceuref{https://mozart.github.io/}
@end itemize

@item Pascal
@itemize @bullet
@item
GNU Pascal Compiler @spaceuref{http://www.gnu-pascal.de/} @* GMP unit.
@item
Numerix @spaceuref{http://pauillac.inria.fr/~quercia/} @* For Free Pascal,
optionally using GMP.
@end itemize

@item Perl
@itemize @bullet
@item
GMP module, see @file{demos/perl} in the GMP sources (@pxref{Demonstration
Programs}).
@item
Math::GMP @spaceuref{https://www.cpan.org/} @* Compatible with Math::BigInt, but
not as many functions as the GMP module above.
@item
Math::BigInt::GMP @spaceuref{https://www.cpan.org/} @* Plug Math::GMP into
normal Math::BigInt operations.
@end itemize

@need 1000
@item Pike
@itemize @bullet
@item
pikempz module in the standard distribution, @uref{https://pike.lysator.liu.se/}
@end itemize

@need 500
@item Prolog
@itemize @bullet
@item
SWI Prolog @spaceuref{http://www.swi-prolog.org/} @*
Arbitrary precision floats.
@end itemize

@item Python
@itemize @bullet
@item
GMPY @uref{https://code.google.com/p/gmpy/}
@end itemize

@item Ruby
@itemize @bullet
@item
@uref{https://rubygems.org/gems/gmp}
@end itemize

@item Scheme
@itemize @bullet
@item
GNU Guile @spaceuref{https://www.gnu.org/software/guile/guile.html}
@item
RScheme @spaceuref{https://www.rscheme.org/}
@item
STklos @spaceuref{http://www.stklos.net/}
@c
@c  For reference, MzScheme uses some of gmp, but (as of version 205) it only
@c  has copies of some of the generic C code, and we don't consider that a
@c  language binding to gmp.
@c
@end itemize

@item Smalltalk
@itemize @bullet
@item
GNU Smalltalk @spaceuref{http://smalltalk.gnu.org/}
@end itemize

@item Other
@itemize @bullet
@item
Axiom @uref{https://savannah.nongnu.org/projects/axiom} @* Computer algebra
using GCL.
@item
DrGenius @spaceuref{http://drgenius.seul.org/} @* Geometry system and
mathematical programming language.
@item
GiNaC @spaceuref{httsp://www.ginac.de/} @* C++ computer algebra using CLN.
@item
GOO @spaceuref{https://www.eecs.berkeley.edu/~jrb/goo/} @* Dynamic object oriented
language.
@item
Maxima @uref{https://www.ma.utexas.edu/users/wfs/maxima.html} @* Macsyma
computer algebra using GCL.
@c @item
@c Q @spaceuref{http://q-lang.sourceforge.net/} @* Equational programming system.
@item
Regina @spaceuref{http://regina.sourceforge.net/} @* Topological calculator.
@item
Yacas @spaceuref{http://yacas.sourceforge.net} @* Yet another computer algebra system.
@end itemize

@end table


@node Algorithms, Internals, Language Bindings, Top
@chapter Algorithms
@cindex Algorithms

This chapter is an introduction to some of the algorithms used for various GMP
operations.  The code is likely to be hard to understand without knowing
something about the algorithms.

Some GMP internals are mentioned, but applications that expect to be
compatible with future GMP releases should take care to use only the
documented functions.

@menu
* Multiplication Algorithms::
* Division Algorithms::
* Greatest Common Divisor Algorithms::
* Powering Algorithms::
* Root Extraction Algorithms::
* Radix Conversion Algorithms::
* Other Algorithms::
* Assembly Coding::
@end menu


@node Multiplication Algorithms, Division Algorithms, Algorithms, Algorithms
@section Multiplication
@cindex Multiplication algorithms

N@cross{}N limb multiplications and squares are done using one of seven
algorithms, as the size N increases.

@quotation
@multitable {KaratsubaMMM} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item Algorithm @tab Threshold
@item Basecase  @tab (none)
@item Karatsuba @tab @code{MUL_TOOM22_THRESHOLD}
@item Toom-3    @tab @code{MUL_TOOM33_THRESHOLD}
@item Toom-4    @tab @code{MUL_TOOM44_THRESHOLD}
@item Toom-6.5  @tab @code{MUL_TOOM6H_THRESHOLD}
@item Toom-8.5  @tab @code{MUL_TOOM8H_THRESHOLD}
@item FFT       @tab @code{MUL_FFT_THRESHOLD}
@end multitable
@end quotation

Similarly for squaring, with the @code{SQR} thresholds.

N@cross{}M multiplications of operands with different sizes above
@code{MUL_TOOM22_THRESHOLD} are currently done by special Toom-inspired
algorithms or directly with FFT, depending on operand size (@pxref{Unbalanced
Multiplication}).

@menu
* Basecase Multiplication::
* Karatsuba Multiplication::
* Toom 3-Way Multiplication::
* Toom 4-Way Multiplication::
* Higher degree Toom'n'half::
* FFT Multiplication::
* Other Multiplication::
* Unbalanced Multiplication::
@end menu


@node Basecase Multiplication, Karatsuba Multiplication, Multiplication Algorithms, Multiplication Algorithms
@subsection Basecase Multiplication

Basecase N@cross{}M multiplication is a straightforward rectangular set of
cross-products, the same as long multiplication done by hand and for that
reason sometimes known as the schoolbook or grammar school method.  This is an
@m{O(NM),O(N*M)} algorithm.  See Knuth section 4.3.1 algorithm M
(@pxref{References}), and the @file{mpn/generic/mul_basecase.c} code.

Assembly implementations of @code{mpn_mul_basecase} are essentially the same
as the generic C code, but have all the usual assembly tricks and
obscurities introduced for speed.

A square can be done in roughly half the time of a multiply, by using the fact
that the cross products above and below the diagonal are the same.  A triangle
of products below the diagonal is formed, doubled (left shift by one bit), and
then the products on the diagonal added.  This can be seen in
@file{mpn/generic/sqr_basecase.c}.  Again the assembly implementations take
essentially the same approach.

@tex
\def\GMPline#1#2#3#4#5#6{%
  \hbox {%
    \vrule height 2.5ex depth 1ex
           \hbox to 2em {\hfil{#2}\hfil}%
    \vrule \hbox to 2em {\hfil{#3}\hfil}%
    \vrule \hbox to 2em {\hfil{#4}\hfil}%
    \vrule \hbox to 2em {\hfil{#5}\hfil}%
    \vrule \hbox to 2em {\hfil{#6}\hfil}%
    \vrule}}
\GMPdisplay{
  \hbox{%
    \vbox{%
      \hbox to 1.5em {\vrule height 2.5ex depth 1ex width 0pt}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u0\hfil}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u1\hfil}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u2\hfil}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u3\hfil}%
      \hbox {\vrule height 2.5ex depth 1ex width 0pt u4\hfil}%
      \vfill}%
    \vbox{%
      \hbox{%
        \hbox to 2em {\hfil u0\hfil}%
        \hbox to 2em {\hfil u1\hfil}%
        \hbox to 2em {\hfil u2\hfil}%
        \hbox to 2em {\hfil u3\hfil}%
        \hbox to 2em {\hfil u4\hfil}}%
      \vskip 0.7ex
      \hrule
      \GMPline{u0}{d}{}{}{}{}%
      \hrule
      \GMPline{u1}{}{d}{}{}{}%
      \hrule
      \GMPline{u2}{}{}{d}{}{}%
      \hrule
      \GMPline{u3}{}{}{}{d}{}%
      \hrule
      \GMPline{u4}{}{}{}{}{d}%
      \hrule}}}
@end tex
@ifnottex
@example
@group
     u0  u1  u2  u3  u4
   +---+---+---+---+---+
u0 | d |   |   |   |   |
   +---+---+---+---+---+
u1 |   | d |   |   |   |
   +---+---+---+---+---+
u2 |   |   | d |   |   |
   +---+---+---+---+---+
u3 |   |   |   | d |   |
   +---+---+---+---+---+
u4 |   |   |   |   | d |
   +---+---+---+---+---+
@end group
@end example
@end ifnottex

In practice squaring isn't a full 2@cross{} faster than multiplying, it's
usually around 1.5@cross{}.  Less than 1.5@cross{} probably indicates
@code{mpn_sqr_basecase} wants improving on that CPU.

On some CPUs @code{mpn_mul_basecase} can be faster than the generic C
@code{mpn_sqr_basecase} on some small sizes.  @code{SQR_BASECASE_THRESHOLD} is
the size at which to use @code{mpn_sqr_basecase}, this will be zero if that
routine should be used always.


@node Karatsuba Multiplication, Toom 3-Way Multiplication, Basecase Multiplication, Multiplication Algorithms
@subsection Karatsuba Multiplication
@cindex Karatsuba multiplication

The Karatsuba multiplication algorithm is described in Knuth section 4.3.3
part A, and various other textbooks.  A brief description is given here.

The inputs @math{x} and @math{y} are treated as each split into two parts of
equal length (or the most significant part one limb shorter if N is odd).

@tex
% GMPboxwidth used for all the multiplication pictures
\global\newdimen\GMPboxwidth \global\GMPboxwidth=5em
% GMPboxdepth and GMPboxheight are also used for the float pictures
\global\newdimen\GMPboxdepth  \global\GMPboxdepth=1ex
\global\newdimen\GMPboxheight \global\GMPboxheight=2ex
\gdef\GMPvrule{\vrule height \GMPboxheight depth \GMPboxdepth}
\def\GMPbox#1#2{%
  \vbox {%
    \hrule
    \hbox to 2\GMPboxwidth{%
      \GMPvrule \hfil $#1$\hfil \vrule \hfil $#2$\hfil \vrule}%
    \hrule}}
\GMPdisplay{%
\vbox{%
  \hbox to 2\GMPboxwidth {high \hfil low}
  \vskip 0.7ex
  \GMPbox{x_1}{x_0}
  \vskip 0.5ex
  \GMPbox{y_1}{y_0}
}}
@end tex
@ifnottex
@example
@group
 high              low
+----------+----------+
|    x1    |    x0    |
+----------+----------+

+----------+----------+
|    y1    |    y0    |
+----------+----------+
@end group
@end example
@end ifnottex

Let @math{b} be the power of 2 where the split occurs, i.e.@: if @ms{x,0} is
@math{k} limbs (@ms{y,0} the same) then
@m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}.
With that @m{x=x_1b+x_0,x=x1*b+x0} and @m{y=y_1b+y_0,y=y1*b+y0}, and the
following holds,

@display
@m{xy = (b^2+b)x_1y_1 - b(x_1-x_0)(y_1-y_0) + (b+1)x_0y_0,
  x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0}
@end display

This formula means doing only three multiplies of (N/2)@cross{}(N/2) limbs,
whereas a basecase multiply of N@cross{}N limbs is equivalent to four
multiplies of (N/2)@cross{}(N/2).  The factors @math{(b^2+b)} etc represent
the positions where the three products must be added.

@tex
\def\GMPboxA#1#2{%
  \vbox{%
    \hrule
    \hbox{%
      \GMPvrule
      \hbox to 2\GMPboxwidth {\hfil\hbox{$#1$}\hfil}%
      \vrule
      \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
      \vrule}
    \hrule}}
\def\GMPboxB#1#2{%
  \hbox{%
    \raise \GMPboxdepth \hbox to \GMPboxwidth {\hfil #1\hskip 0.5em}%
    \vbox{%
      \hrule
      \hbox{%
        \GMPvrule
        \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}%
        \vrule}%
      \hrule}}}
\GMPdisplay{%
\vbox{%
  \hbox to 4\GMPboxwidth {high \hfil low}
  \vskip 0.7ex
  \GMPboxA{x_1y_1}{x_0y_0}
  \vskip 0.5ex
  \GMPboxB{$+$}{x_1y_1}
  \vskip 0.5ex
  \GMPboxB{$+$}{x_0y_0}
  \vskip 0.5ex
  \GMPboxB{$-$}{(x_1-x_0)(y_1-y_0)}
}}
@end tex
@ifnottex
@example
@group
 high                              low
+--------+--------+ +--------+--------+
|      x1*y1      | |      x0*y0      |
+--------+--------+ +--------+--------+
          +--------+--------+
      add |      x1*y1      |
          +--------+--------+
          +--------+--------+
      add |      x0*y0      |
          +--------+--------+
          +--------+--------+
      sub | (x1-x0)*(y1-y0) |
          +--------+--------+
@end group
@end example
@end ifnottex

The term @m{(x_1-x_0)(y_1-y_0),(x1-x0)*(y1-y0)} is best calculated as an
absolute value, and the sign used to choose to add or subtract.  Notice the
sum @m{\mathop{\rm high}(x_0y_0)+\mathop{\rm low}(x_1y_1),
high(x0*y0)+low(x1*y1)} occurs twice, so it's possible to do @m{5k,5*k} limb
additions, rather than @m{6k,6*k}, but in GMP extra function call overheads
outweigh the saving.

Squaring is similar to multiplying, but with @math{x=y} the formula reduces to
an equivalent with three squares,

@display
@m{x^2 = (b^2+b)x_1^2 - b(x_1-x_0)^2 + (b+1)x_0^2,
   x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2}
@end display

The final result is accumulated from those three squares the same way as for
the three multiplies above.  The middle term @m{(x_1-x_0)^2,(x1-x0)^2} is now
always positive.

A similar formula for both multiplying and squaring can be constructed with a
middle term @m{(x_1+x_0)(y_1+y_0),(x1+x0)*(y1+y0)}.  But those sums can exceed
@math{k} limbs, leading to more carry handling and additions than the form
above.

Karatsuba multiplication is asymptotically an @math{O(N^@W{1.585})} algorithm,
the exponent being @m{\log3/\log2,log(3)/log(2)}, representing 3 multiplies
each @math{1/2} the size of the inputs.  This is a big improvement over the
basecase multiply at @math{O(N^2)} and the advantage soon overcomes the extra
additions Karatsuba performs.  @code{MUL_TOOM22_THRESHOLD} can be as little
as 10 limbs.  The @code{SQR} threshold is usually about twice the @code{MUL}.

The basecase algorithm will take a time of the form @m{M(N) = aN^2 + bN + c,
M(N) = a*N^2 + b*N + c} and the Karatsuba algorithm @m{K(N) = 3M(N/2) + dN +
e, K(N) = 3*M(N/2) + d*N + e}, which expands to @m{K(N) = {3\over4} aN^2 +
{3\over2} bN + 3c + dN + e, K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e}.  The
factor @m{3\over4, 3/4} for @math{a} means per-crossproduct speedups in the
basecase code will increase the threshold since they benefit @math{M(N)} more
than @math{K(N)}.  And conversely the @m{3\over2, 3/2} for @math{b} means
linear style speedups of @math{b} will increase the threshold since they
benefit @math{K(N)} more than @math{M(N)}.  The latter can be seen for
instance when adding an optimized @code{mpn_sqr_diagonal} to
@code{mpn_sqr_basecase}.  Of course all speedups reduce total time, and in
that sense the algorithm thresholds are merely of academic interest.


@node Toom 3-Way Multiplication, Toom 4-Way Multiplication, Karatsuba Multiplication, Multiplication Algorithms
@subsection Toom 3-Way Multiplication
@cindex Toom multiplication

The Karatsuba formula is the simplest case of a general approach to splitting
inputs that leads to both Toom and FFT algorithms.  A description of
Toom can be found in Knuth section 4.3.3, with an example 3-way
calculation after Theorem A@.  The 3-way form used in GMP is described here.

The operands are each considered split into 3 pieces of equal length (or the
most significant part 1 or 2 limbs shorter than the other two).

@tex
\def\GMPbox#1#2#3{%
  \vbox{%
    \hrule \vfil
    \hbox to 3\GMPboxwidth {%
      \GMPvrule
      \hfil$#1$\hfil
      \vrule
      \hfil$#2$\hfil
      \vrule
      \hfil$#3$\hfil
      \vrule}%
    \vfil \hrule
}}
\GMPdisplay{%
\vbox{%
  \hbox to 3\GMPboxwidth {high \hfil low}
  \vskip 0.7ex
  \GMPbox{x_2}{x_1}{x_0}
  \vskip 0.5ex
  \GMPbox{y_2}{y_1}{y_0}
  \vskip 0.5ex
}}
@end tex
@ifnottex
@example
@group
 high                         low
+----------+----------+----------+
|    x2    |    x1    |    x0    |
+----------+----------+----------+

+----------+----------+----------+
|    y2    |    y1    |    y0    |
+----------+----------+----------+
@end group
@end example
@end ifnottex

@noindent
These parts are treated as the coefficients of two polynomials

@display
@group
@m{X(t) = x_2t^2 + x_1t + x_0,
   X(t) = x2*t^2 + x1*t + x0}
@m{Y(t) = y_2t^2 + y_1t + y_0,
   Y(t) = y2*t^2 + y1*t + y0}
@end group
@end display

Let @math{b} equal the power of 2 which is the size of the @ms{x,0}, @ms{x,1},
@ms{y,0} and @ms{y,1} pieces, i.e.@: if they're @math{k} limbs each then
@m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}.
With this @math{x=X(b)} and @math{y=Y(b)}.

Let a polynomial @m{W(t)=X(t)Y(t),W(t)=X(t)*Y(t)} and suppose its coefficients
are

@display
@m{W(t) = w_4t^4 + w_3t^3 + w_2t^2 + w_1t + w_0,
   W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0}
@end display

The @m{w_i,w[i]} are going to be determined, and when they are they'll give
the final result using @math{w=W(b)}, since
@m{xy=X(b)Y(b),x*y=X(b)*Y(b)=W(b)}.  The coefficients will be roughly
@math{b^2} each, and the final @math{W(b)} will be an addition like,

@tex
\def\GMPbox#1#2{%
  \moveright #1\GMPboxwidth
  \vbox{%
    \hrule
    \hbox{%
      \GMPvrule
      \hbox to 2\GMPboxwidth {\hfil$#2$\hfil}%
      \vrule}%
    \hrule
}}
\GMPdisplay{%
\vbox{%
  \hbox to 6\GMPboxwidth {high \hfil low}%
  \vskip 0.7ex
  \GMPbox{0}{w_4}
  \vskip 0.5ex
  \GMPbox{1}{w_3}
  \vskip 0.5ex
  \GMPbox{2}{w_2}
  \vskip 0.5ex
  \GMPbox{3}{w_1}
  \vskip 0.5ex
  \GMPbox{4}{w_0}
}}
@end tex
@ifnottex
@example
@group
 high                                        low
+-------+-------+
|       w4      |
+-------+-------+
       +--------+-------+
       |        w3      |
       +--------+-------+
               +--------+-------+
               |        w2      |
               +--------+-------+
                       +--------+-------+
                       |        w1      |
                       +--------+-------+
                                +-------+-------+
                                |       w0      |
                                +-------+-------+
@end group
@end example
@end ifnottex

The @m{w_i,w[i]} coefficients could be formed by a simple set of cross
products, like @m{w_4=x_2y_2,w4=x2*y2}, @m{w_3=x_2y_1+x_1y_2,w3=x2*y1+x1*y2},
@m{w_2=x_2y_0+x_1y_1+x_0y_2,w2=x2*y0+x1*y1+x0*y2} etc, but this would need all
nine @m{x_iy_j,x[i]*y[j]} for @math{i,j=0,1,2}, and would be equivalent merely
to a basecase multiply.  Instead the following approach is used.

@math{X(t)} and @math{Y(t)} are evaluated and multiplied at 5 points, giving
values of @math{W(t)} at those points.  In GMP the following points are used,

@quotation
@multitable {@m{t=\infty,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item Point                 @tab Value
@item @math{t=0}            @tab @m{x_0y_0,x0 * y0}, which gives @ms{w,0} immediately
@item @math{t=1}            @tab @m{(x_2+x_1+x_0)(y_2+y_1+y_0),(x2+x1+x0) * (y2+y1+y0)}
@item @math{t=-1}           @tab @m{(x_2-x_1+x_0)(y_2-y_1+y_0),(x2-x1+x0) * (y2-y1+y0)}
@item @math{t=2}            @tab @m{(4x_2+2x_1+x_0)(4y_2+2y_1+y_0),(4*x2+2*x1+x0) * (4*y2+2*y1+y0)}
@item @m{t=\infty,t=inf}    @tab @m{x_2y_2,x2 * y2}, which gives @ms{w,4} immediately
@end multitable
@end quotation

At @math{t=-1} the values can be negative and that's handled using the
absolute values and tracking the sign separately.  At @m{t=\infty,t=inf} the
value is actually @m{\lim_{t\to\infty} {X(t)Y(t)\over t^4}, X(t)*Y(t)/t^4 in
the limit as t approaches infinity}, but it's much easier to think of as
simply @m{x_2y_2,x2*y2} giving @ms{w,4} immediately (much like
@m{x_0y_0,x0*y0} at @math{t=0} gives @ms{w,0} immediately).

Each of the points substituted into
@m{W(t)=w_4t^4+\cdots+w_0,W(t)=w4*t^4+@dots{}+w0} gives a linear combination
of the @m{w_i,w[i]} coefficients, and the value of those combinations has just
been calculated.

@tex
\GMPdisplay{%
$\matrix{%
W(0)      & = &       &   &      &   &      &   &      &   & w_0 \cr
W(1)      & = &   w_4 & + &  w_3 & + &  w_2 & + &  w_1 & + & w_0 \cr
W(-1)     & = &   w_4 & - &  w_3 & + &  w_2 & - &  w_1 & + & w_0 \cr
W(2)      & = & 16w_4 & + & 8w_3 & + & 4w_2 & + & 2w_1 & + & w_0 \cr
W(\infty) & = &   w_4 \cr
}$}
@end tex
@ifnottex
@example
@group
W(0)   =                              w0
W(1)   =    w4 +   w3 +   w2 +   w1 + w0
W(-1)  =    w4 -   w3 +   w2 -   w1 + w0
W(2)   = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
W(inf) =    w4
@end group
@end example
@end ifnottex

This is a set of five equations in five unknowns, and some elementary linear
algebra quickly isolates each @m{w_i,w[i]}.  This involves adding or
subtracting one @math{W(t)} value from another, and a couple of divisions by
powers of 2 and one division by 3, the latter using the special
@code{mpn_divexact_by3} (@pxref{Exact Division}).

The conversion of @math{W(t)} values to the coefficients is interpolation.  A
polynomial of degree 4 like @math{W(t)} is uniquely determined by values known
at 5 different points.  The points are arbitrary and can be chosen to make the
linear equations come out with a convenient set of steps for quickly isolating
the @m{w_i,w[i]}.

Squaring follows the same procedure as multiplication, but there's only one
@math{X(t)} and it's evaluated at the 5 points, and those values squared to
give values of @math{W(t)}.  The interpolation is then identical, and in fact
the same @code{toom_interpolate_5pts} subroutine is used for both squaring and
multiplying.

Toom-3 is asymptotically @math{O(N^@W{1.465})}, the exponent being
@m{\log5/\log3,log(5)/log(3)}, representing 5 recursive multiplies of 1/3 the
original size each.  This is an improvement over Karatsuba at
@math{O(N^@W{1.585})}, though Toom does more work in the evaluation and
interpolation and so it only realizes its advantage above a certain size.

Near the crossover between Toom-3 and Karatsuba there's generally a range of
sizes where the difference between the two is small.
@code{MUL_TOOM33_THRESHOLD} is a somewhat arbitrary point in that range and
successive runs of the tune program can give different values due to small
variations in measuring.  A graph of time versus size for the two shows the
effect, see @file{tune/README}.

At the fairly small sizes where the Toom-3 thresholds occur it's worth
remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be
expected to make accurate predictions, due of course to the big influence of
all sorts of overheads, and the fact that only a few recursions of each are
being performed.  Even at large sizes there's a good chance machine dependent
effects like cache architecture will mean actual performance deviates from
what might be predicted.

The formula given for the Karatsuba algorithm (@pxref{Karatsuba
Multiplication}) has an equivalent for Toom-3 involving only five multiplies,
but this would be complicated and unenlightening.

An alternate view of Toom-3 can be found in Zuras (@pxref{References}), using
a vector to represent the @math{x} and @math{y} splits and a matrix
multiplication for the evaluation and interpolation stages.  The matrix
inverses are not meant to be actually used, and they have elements with values
much greater than in fact arise in the interpolation steps.  The diagram shown
for the 3-way is attractive, but again doesn't have to be implemented that way
and for example with a bit of rearrangement just one division by 6 can be
done.


@node Toom 4-Way Multiplication, Higher degree Toom'n'half, Toom 3-Way Multiplication, Multiplication Algorithms
@subsection Toom 4-Way Multiplication
@cindex Toom multiplication

Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
respectively.  Toom-4 analogously splits the operands into 4 coefficients.
Using the notation from the section on Toom-3 multiplication, we form two
polynomials:

@display
@group
@m{X(t) = x_3t^3 + x_2t^2 + x_1t + x_0,
   X(t) = x3*t^3 + x2*t^2 + x1*t + x0}
@m{Y(t) = y_3t^3 + y_2t^2 + y_1t + y_0,
   Y(t) = y3*t^3 + y2*t^2 + y1*t + y0}
@end group
@end display

@math{X(t)} and @math{Y(t)} are evaluated and multiplied at 7 points, giving
values of @math{W(t)} at those points.  In GMP the following points are used,

@quotation
@multitable {@m{t=-1/2,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item Point              @tab Value
@item @math{t=0}         @tab @m{x_0y_0,x0 * y0}, which gives @ms{w,0} immediately
@item @math{t=1/2}       @tab @m{(x_3+2x_2+4x_1+8x_0)(y_3+2y_2+4y_1+8y_0),(x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)}
@item @math{t=-1/2}      @tab @m{(-x_3+2x_2-4x_1+8x_0)(-y_3+2y_2-4y_1+8y_0),(-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)}
@item @math{t=1}         @tab @m{(x_3+x_2+x_1+x_0)(y_3+y_2+y_1+y_0),(x3+x2+x1+x0) * (y3+y2+y1+y0)}
@item @math{t=-1}        @tab @m{(-x_3+x_2-x_1+x_0)(-y_3+y_2-y_1+y_0),(-x3+x2-x1+x0) * (-y3+y2-y1+y0)}
@item @math{t=2}         @tab @m{(8x_3+4x_2+2x_1+x_0)(8y_3+4y_2+2y_1+y_0),(8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)}
@item @m{t=\infty,t=inf} @tab @m{x_3y_3,x3 * y3}, which gives @ms{w,6} immediately
@end multitable
@end quotation

The number of additions and subtractions for Toom-4 is much larger than for Toom-3.
But several subexpressions occur multiple times, for example @m{x_2+x_0,x2+x0}, occurs
for both @math{t=1} and @math{t=-1}.

Toom-4 is asymptotically @math{O(N^@W{1.404})}, the exponent being
@m{\log7/\log4,log(7)/log(4)}, representing 7 recursive multiplies of 1/4 the
original size each.


@node Higher degree Toom'n'half, FFT Multiplication, Toom 4-Way Multiplication, Multiplication Algorithms
@subsection Higher degree Toom'n'half
@cindex Toom multiplication

The Toom algorithms described above (@pxref{Toom 3-Way Multiplication},
@pxref{Toom 4-Way Multiplication}) generalizes to split into an arbitrary
number of pieces. In general a split of two equally long operands into
@math{r} pieces leads to evaluations and pointwise multiplications done at
@m{2r-1,2*r-1} points. To fully exploit symmetries it would be better to have
a multiple of 4 points, that's why for higher degree Toom'n'half is used.

Toom'n'half means that the existence of one more piece is considered for a
single operand. It can be virtual, i.e. zero, or real, when the two operand
are not exactly balanced. By choosing an even @math{r},
Toom-@m{r{1\over2},r+1/2} requires @math{2r} points, a multiple of four.

The quadruplets of points include 0, @m{\infty,inf}, +1, -1 and
@m{\pm2^i,+-2^i}, @m{\pm2^{-i},+-2^-i} . Each of them giving shortcuts for the
evaluation phase and for some steps in the interpolation phase. Further tricks
are used to reduce the memory footprint of the whole multiplication algorithm
to a memory buffer equal in size to the result of the product.

Current GMP uses both Toom-6'n'half and Toom-8'n'half.


@node FFT Multiplication, Other Multiplication, Higher degree Toom'n'half, Multiplication Algorithms
@subsection FFT Multiplication
@cindex FFT multiplication
@cindex Fast Fourier Transform

At large to very large sizes a Fermat style FFT multiplication is used,
following Sch@"onhage and Strassen (@pxref{References}).  Descriptions of FFTs
in various forms can be found in many textbooks, for instance Knuth section
4.3.3 part C or Lipson chapter IX@.  A brief description of the form used in
GMP is given here.

The multiplication done is @m{xy \bmod 2^N+1, x*y mod 2^N+1}, for a given
@math{N}.  A full product @m{xy,x*y} is obtained by choosing @m{N \ge
\mathop{\rm bits}(x)+\mathop{\rm bits}(y), N>=bits(x)+bits(y)} and padding
@math{x} and @math{y} with high zero limbs.  The modular product is the native
form for the algorithm, so padding to get a full product is unavoidable.

The algorithm follows a split, evaluate, pointwise multiply, interpolate and
combine similar to that described above for Karatsuba and Toom-3.  A @math{k}
parameter controls the split, with an FFT-@math{k} splitting into @math{2^k}
pieces of @math{M=N/2^k} bits each.  @math{N} must be a multiple of
@m{2^k\times@code{mp\_bits\_per\_limb}, (2^k)*@nicode{mp_bits_per_limb}} so
the split falls on limb boundaries, avoiding bit shifts in the split and
combine stages.

The evaluations, pointwise multiplications, and interpolation, are all done
modulo @m{2^{N'}+1, 2^N'+1} where @math{N'} is @math{2M+k+3} rounded up to a
multiple of @math{2^k} and of @code{mp_bits_per_limb}.  The results of
interpolation will be the following negacyclic convolution of the input
pieces, and the choice of @math{N'} ensures these sums aren't truncated.
@tex
$$ w_n = \sum_{{i+j = b2^k+n}\atop{b=0,1}} (-1)^b x_i y_j $$
@end tex
@ifnottex

@example
           ---
           \         b
w[n] =     /     (-1) * x[i] * y[j]
           ---
       i+j==b*2^k+n
          b=0,1
@end example

@end ifnottex
The points used for the evaluation are @math{g^i} for @math{i=0} to
@math{2^k-1} where @m{g=2^{2N'/2^k}, g=2^(2N'/2^k)}.  @math{g} is a
@m{2^k,2^k'}th root of unity mod @m{2^{N'}+1,2^N'+1}, which produces necessary
cancellations at the interpolation stage, and it's also a power of 2 so the
fast Fourier transforms used for the evaluation and interpolation do only
shifts, adds and negations.

The pointwise multiplications are done modulo @m{2^{N'}+1, 2^N'+1} and either
recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or
basecase), whichever is optimal at the size @math{N'}.  The interpolation is
an inverse fast Fourier transform.  The resulting set of sums of @m{x_iy_j,
x[i]*y[j]} are added at appropriate offsets to give the final result.

Squaring is the same, but @math{x} is the only input so it's one transform at
the evaluate stage and the pointwise multiplies are squares.  The
interpolation is the same.

For a mod @math{2^N+1} product, an FFT-@math{k} is an @m{O(N^{k/(k-1)}),
O(N^(k/(k-1)))} algorithm, the exponent representing @math{2^k} recursed
modular multiplies each @m{1/2^{k-1},1/2^(k-1)} the size of the original.
Each successive @math{k} is an asymptotic improvement, but overheads mean each
is only faster at bigger and bigger sizes.  In the code, @code{MUL_FFT_TABLE}
and @code{SQR_FFT_TABLE} are the thresholds where each @math{k} is used.  Each
new @math{k} effectively swaps some multiplying for some shifts, adds and
overheads.

A mod @math{2^N+1} product can be formed with a normal
@math{N@cross{}N@rightarrow{}2N} bit multiply plus a subtraction, so an FFT
and Toom-3 etc can be compared directly.  A @math{k=4} FFT at
@math{O(N^@W{1.333})} can be expected to be the first faster than Toom-3 at
@math{O(N^@W{1.465})}.  In practice this is what's found, with
@code{MUL_FFT_MODF_THRESHOLD} and @code{SQR_FFT_MODF_THRESHOLD} being between
300 and 1000 limbs, depending on the CPU@.  So far it's been found that only
very large FFTs recurse into pointwise multiplies above these sizes.

When an FFT is to give a full product, the change of @math{N} to @math{2N}
doesn't alter the theoretical complexity for a given @math{k}, but for the
purposes of considering where an FFT might be first used it can be assumed
that the FFT is recursing into a normal multiply and that on that basis it's
doing @math{2^k} recursed multiplies each @m{1/2^{k-2},1/2^(k-2)} the size of
the inputs, making it @m{O(N^{k/(k-2)}), O(N^(k/(k-2)))}.  This would mean
@math{k=7} at @math{O(N^@W{1.4})} would be the first FFT faster than Toom-3.
In practice @code{MUL_FFT_THRESHOLD} and @code{SQR_FFT_THRESHOLD} have been
found to be in the @math{k=8} range, somewhere between 3000 and 10000 limbs.

The way @math{N} is split into @math{2^k} pieces and then @math{2M+k+3} is
rounded up to a multiple of @math{2^k} and @code{mp_bits_per_limb} means that
when @math{2^k@ge{}@nicode{mp\_bits\_per\_limb}} the effective @math{N} is a
multiple of @m{2^{2k-1},2^(2k-1)} bits.  The @math{+k+3} means some values of
@math{N} just under such a multiple will be rounded to the next.  The
complexity calculations above assume that a favourable size is used, meaning
one which isn't padded through rounding, and it's also assumed that the extra
@math{+k+3} bits are negligible at typical FFT sizes.

The practical effect of the @m{2^{2k-1},2^(2k-1)} constraint is to introduce a
step-effect into measured speeds.  For example @math{k=8} will round @math{N}
up to a multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
groups of sizes for which @code{mpn_mul_n} runs at the same speed.  Or for
@math{k=9} groups of 2048 limbs, @math{k=10} groups of 8192 limbs, etc.  In
practice it's been found each @math{k} is used at quite small multiples of its
size constraint and so the step effect is quite noticeable in a time versus
size graph.

The threshold determinations currently measure at the mid-points of size
steps, but this is sub-optimal since at the start of a new step it can happen
that it's better to go back to the previous @math{k} for a while.  Something
more sophisticated for @code{MUL_FFT_TABLE} and @code{SQR_FFT_TABLE} will be
needed.


@node Other Multiplication, Unbalanced Multiplication, FFT Multiplication, Multiplication Algorithms
@subsection Other Multiplication
@cindex Toom multiplication

The Toom algorithms described above (@pxref{Toom 3-Way Multiplication},
@pxref{Toom 4-Way Multiplication}) generalizes to split into an arbitrary
number of pieces, as per Knuth section 4.3.3 algorithm C@.  This is not
currently used.  The notes here are merely for interest.

In general a split into @math{r+1} pieces is made, and evaluations and
pointwise multiplications done at @m{2r+1,2*r+1} points.  A 4-way split does 7
pointwise multiplies, 5-way does 9, etc.  Asymptotically an @math{(r+1)}-way
algorithm is @m{O(N^{log(2r+1)/log(r+1)}), O(N^(log(2*r+1)/log(r+1)))}.  Only
the pointwise multiplications count towards big-@math{O} complexity, but the
time spent in the evaluate and interpolate stages grows with @math{r} and has
a significant practical impact, with the asymptotic advantage of each @math{r}
realized only at bigger and bigger sizes.  The overheads grow as
@m{O(Nr),O(N*r)}, whereas in an @math{r=2^k} FFT they grow only as @m{O(N \log
r), O(N*log(r))}.

Knuth algorithm C evaluates at points 0,1,2,@dots{},@m{2r,2*r}, but exercise 4
uses @math{-r},@dots{},0,@dots{},@math{r} and the latter saves some small
multiplies in the evaluate stage (or rather trades them for additions), and
has a further saving of nearly half the interpolate steps.  The idea is to
separate odd and even final coefficients and then perform algorithm C steps C7
and C8 on them separately.  The divisors at step C7 become @math{j^2} and the
multipliers at C8 become @m{2tj-j^2,2*t*j-j^2}.

Splitting odd and even parts through positive and negative points can be
thought of as using @math{-1} as a square root of unity.  If a 4th root of
unity was available then a further split and speedup would be possible, but no
such root exists for plain integers.  Going to complex integers with
@m{i=\sqrt{-1}, i=sqrt(-1)} doesn't help, essentially because in Cartesian
form it takes three real multiplies to do a complex multiply.  The existence
of @m{2^k,2^k'}th roots of unity in a suitable ring or field lets the fast
Fourier transform keep splitting and get to @m{O(N \log r), O(N*log(r))}.

Floating point FFTs use complex numbers approximating Nth roots of unity.
Some processors have special support for such FFTs.  But these are not used in
GMP since it's very difficult to guarantee an exact result (to some number of
bits).  An occasional difference of 1 in the last bit might not matter to a
typical signal processing algorithm, but is of course of vital importance to
GMP.


@node Unbalanced Multiplication,  , Other Multiplication, Multiplication Algorithms
@subsection Unbalanced Multiplication
@cindex Unbalanced multiplication

Multiplication of operands with different sizes, both below
@code{MUL_TOOM22_THRESHOLD} are done with plain schoolbook multiplication
(@pxref{Basecase Multiplication}).

For really large operands, we invoke FFT directly.

For operands between these sizes, we use Toom inspired algorithms suggested by
Alberto Zanoni and Marco Bodrato.  The idea is to split the operands into
polynomials of different degree.  GMP currently splits the smaller operand
onto 2 coefficients, i.e., a polynomial of degree 1, but the larger operand
can be split into 2, 3, or 4 coefficients, i.e., a polynomial of degree 1 to
3.

@c FIXME: This is mighty ugly, but a cleaner @need triggers texinfo bugs that
@c screws up layout here and there in the rest of the manual.
@c @tex
@c \goodbreak
@c @end tex
@node Division Algorithms, Greatest Common Divisor Algorithms, Multiplication Algorithms, Algorithms
@section Division Algorithms
@cindex Division algorithms

@menu
* Single Limb Division::
* Basecase Division::
* Divide and Conquer Division::
* Block-Wise Barrett Division::
* Exact Division::
* Exact Remainder::
* Small Quotient Division::
@end menu


@node Single Limb Division, Basecase Division, Division Algorithms, Division Algorithms
@subsection Single Limb Division

N@cross{}1 division is implemented using repeated 2@cross{}1 divisions from
high to low, either with a hardware divide instruction or a multiplication by
inverse, whichever is best on a given CPU.

The multiply by inverse follows ``Improved division by invariant integers'' by
M@"oller and Granlund (@pxref{References}) and is implemented as
@code{udiv_qrnnd_preinv} in @file{gmp-impl.h}.  The idea is to have a
fixed-point approximation to @math{1/d} (see @code{invert_limb}) and then
multiply by the high limb (plus one bit) of the dividend to get a quotient
@math{q}.  With @math{d} normalized (high bit set), @math{q} is no more than 1
too small.  Subtracting @m{qd,q*d} from the dividend gives a remainder, and
reveals whether @math{q} or @math{q-1} is correct.

The result is a division done with two multiplications and four or five
arithmetic operations.  On CPUs with low latency multipliers this can be much
faster than a hardware divide, though the cost of calculating the inverse at
the start may mean it's only better on inputs bigger than say 4 or 5 limbs.

When a divisor must be normalized, either for the generic C
@code{__udiv_qrnnd_c} or the multiply by inverse, the division performed is
actually @m{a2^k,a*2^k} by @m{d2^k,d*2^k} where @math{a} is the dividend and
@math{k} is the power necessary to have the high bit of @m{d2^k,d*2^k} set.
The bit shifts for the dividend are usually accomplished ``on the fly''
meaning by extracting the appropriate bits at each step.  Done this way the
quotient limbs come out aligned ready to store.  When only the remainder is
wanted, an alternative is to take the dividend limbs unshifted and calculate
@m{r = a \bmod d2^k, r = a mod d*2^k} followed by an extra final step @m{r2^k
\bmod d2^k, r*2^k mod d*2^k}.  This can help on CPUs with poor bit shifts or
few registers.

The multiply by inverse can be done two limbs at a time.  The calculation is
basically the same, but the inverse is two limbs and the divisor treated as if
padded with a low zero limb.  This means more work, since the inverse will
need a 2@cross{}2 multiply, but the four 1@cross{}1s to do that are
independent and can therefore be done partly or wholly in parallel.  Likewise
for a 2@cross{}1 calculating @m{qd,q*d}.  The net effect is to process two
limbs with roughly the same two multiplies worth of latency that one limb at a
time gives.  This extends to 3 or 4 limbs at a time, though the extra work to
apply the inverse will almost certainly soon reach the limits of multiplier
throughput.

A similar approach in reverse can be taken to process just half a limb at a
time if the divisor is only a half limb.  In this case the 1@cross{}1 multiply
for the inverse effectively becomes two @m{{1\over2}\times1, (1/2)x1} for each
limb, which can be a saving on CPUs with a fast half limb multiply, or in fact
if the only multiply is a half limb, and especially if it's not pipelined.


@node Basecase Division, Divide and Conquer Division, Single Limb Division, Division Algorithms
@subsection Basecase Division

Basecase N@cross{}M division is like long division done by hand, but in base
@m{2\GMPraise{@code{mp\_bits\_per\_limb}}, 2^mp_bits_per_limb}.  See Knuth
section 4.3.1 algorithm D, and @file{mpn/generic/sb_divrem_mn.c}.

Briefly stated, while the dividend remains larger than the divisor, a high
quotient limb is formed and the N@cross{}1 product @m{qd,q*d} subtracted at
the top end of the dividend.  With a normalized divisor (most significant bit
set), each quotient limb can be formed with a 2@cross{}1 division and a
1@cross{}1 multiplication plus some subtractions.  The 2@cross{}1 division is
by the high limb of the divisor and is done either with a hardware divide or a
multiply by inverse (the same as in @ref{Single Limb Division}) whichever is
faster.  Such a quotient is sometimes one too big, requiring an addback of the
divisor, but that happens rarely.

With Q=N@minus{}M being the number of quotient limbs, this is an
@m{O(QM),O(Q*M)} algorithm and will run at a speed similar to a basecase
Q@cross{}M multiplication, differing in fact only in the extra multiply and
divide for each of the Q quotient limbs.


@node Divide and Conquer Division, Block-Wise Barrett Division, Basecase Division, Division Algorithms
@subsection Divide and Conquer Division

For divisors larger than @code{DC_DIV_QR_THRESHOLD}, division is done by dividing.
Or to be precise by a recursive divide and conquer algorithm based on work by
Moenck and Borodin, Jebelean, and Burnikel and Ziegler (@pxref{References}).

The algorithm consists essentially of recognising that a 2N@cross{}N division
can be done with the basecase division algorithm (@pxref{Basecase Division}),
but using N/2 limbs as a base, not just a single limb.  This way the
multiplications that arise are (N/2)@cross{}(N/2) and can take advantage of
Karatsuba and higher multiplication algorithms (@pxref{Multiplication
Algorithms}).  The two ``digits'' of the quotient are formed by recursive
N@cross{}(N/2) divisions.

If the (N/2)@cross{}(N/2) multiplies are done with a basecase multiplication
then the work is about the same as a basecase division, but with more function
call overheads and with some subtractions separated from the multiplies.
These overheads mean that it's only when N/2 is above
@code{MUL_TOOM22_THRESHOLD} that divide and conquer is of use.

@code{DC_DIV_QR_THRESHOLD} is based on the divisor size N, so it will be somewhere
above twice @code{MUL_TOOM22_THRESHOLD}, but how much above depends on the
CPU@.  An optimized @code{mpn_mul_basecase} can lower @code{DC_DIV_QR_THRESHOLD} a
little by offering a ready-made advantage over repeated @code{mpn_submul_1}
calls.

Divide and conquer is asymptotically @m{O(M(N)\log N),O(M(N)*log(N))} where
@math{M(N)} is the time for an N@cross{}N multiplication done with FFTs.  The
actual time is a sum over multiplications of the recursed sizes, as can be
seen near the end of section 2.2 of Burnikel and Ziegler.  For example, within
the Toom-3 range, divide and conquer is @m{2.63M(N), 2.63*M(N)}.  With higher
algorithms the @math{M(N)} term improves and the multiplier tends to @m{\log
N, log(N)}.  In practice, at moderate to large sizes, a 2N@cross{}N division
is about 2 to 4 times slower than an N@cross{}N multiplication.


@node Block-Wise Barrett Division, Exact Division, Divide and Conquer Division, Division Algorithms
@subsection Block-Wise Barrett Division

For the largest divisions, a block-wise Barrett division algorithm is used.
Here, the divisor is inverted to a precision determined by the relative size of
the dividend and divisor.  Blocks of quotient limbs are then generated by
multiplying blocks from the dividend by the inverse.

Our block-wise algorithm computes a smaller inverse than in the plain Barrett
algorithm.  For a @math{2n/n} division, the inverse will be just @m{\lceil n/2
\rceil, ceil(n/2)} limbs.


@node Exact Division, Exact Remainder, Block-Wise Barrett Division, Division Algorithms
@subsection Exact Division


A so-called exact division is when the dividend is known to be an exact
multiple of the divisor.  Jebelean's exact division algorithm uses this
knowledge to make some significant optimizations (@pxref{References}).

The idea can be illustrated in decimal for example with 368154 divided by
543.  Because the low digit of the dividend is 4, the low digit of the
quotient must be 8.  This is arrived at from @m{4 \mathord{\times} 7 \bmod 10,
4*7 mod 10}, using the fact 7 is the modular inverse of 3 (the low digit of
the divisor), since @m{3 \mathord{\times} 7 \mathop{\equiv} 1 \bmod 10, 3*7
@equiv{} 1 mod 10}.  So @m{8\mathord{\times}543 = 4344,8*543=4344} can be
subtracted from the dividend leaving 363810.  Notice the low digit has become
zero.

The procedure is repeated at the second digit, with the next quotient digit 7
(@m{1 \mathord{\times} 7 \bmod 10, 7 @equiv{} 1*7 mod 10}), subtracting
@m{7\mathord{\times}543 = 3801,7*543=3801}, leaving 325800.  And finally at
the third digit with quotient digit 6 (@m{8 \mathord{\times} 7 \bmod 10, 8*7
mod 10}), subtracting @m{6\mathord{\times}543 = 3258,6*543=3258} leaving 0.
So the quotient is 678.

Notice however that the multiplies and subtractions don't need to extend past
the low three digits of the dividend, since that's enough to determine the
three quotient digits.  For the last quotient digit no subtraction is needed
at all.  On a 2N@cross{}N division like this one, only about half the work of
a normal basecase division is necessary.

For an N@cross{}M exact division producing Q=N@minus{}M quotient limbs, the
saving over a normal basecase division is in two parts.  Firstly, each of the
Q quotient limbs needs only one multiply, not a 2@cross{}1 divide and
multiply.  Secondly, the crossproducts are reduced when @math{Q>M} to
@m{QM-M(M+1)/2,Q*M-M*(M+1)/2}, or when @math{Q@le{}M} to @m{Q(Q-1)/2,
Q*(Q-1)/2}.  Notice the savings are complementary.  If Q is big then many
divisions are saved, or if Q is small then the crossproducts reduce to a small
number.

The modular inverse used is calculated efficiently by @code{binvert_limb} in
@file{gmp-impl.h}.  This does four multiplies for a 32-bit limb, or six for a
64-bit limb.  @file{tune/modlinv.c} has some alternate implementations that
might suit processors better at bit twiddling than multiplying.

The sub-quadratic exact division described by Jebelean in ``Exact Division
with Karatsuba Complexity'' is not currently implemented.  It uses a
rearrangement similar to the divide and conquer for normal division
(@pxref{Divide and Conquer Division}), but operating from low to high.  A
further possibility not currently implemented is ``Bidirectional Exact Integer
Division'' by Krandick and Jebelean which forms quotient limbs from both the
high and low ends of the dividend, and can halve once more the number of
crossproducts needed in a 2N@cross{}N division.

A special case exact division by 3 exists in @code{mpn_divexact_by3},
supporting Toom-3 multiplication and @code{mpq} canonicalizations.  It forms
quotient digits with a multiply by the modular inverse of 3 (which is
@code{0xAA..AAB}) and uses two comparisons to determine a borrow for the next
limb.  The multiplications don't need to be on the dependent chain, as long as
the effect of the borrows is applied, which can help chips with pipelined
multipliers.


@node Exact Remainder, Small Quotient Division, Exact Division, Division Algorithms
@subsection Exact Remainder
@cindex Exact remainder

If the exact division algorithm is done with a full subtraction at each stage
and the dividend isn't a multiple of the divisor, then low zero limbs are
produced but with a remainder in the high limbs.  For dividend @math{a},
divisor @math{d}, quotient @math{q}, and @m{b = 2
\GMPraise{@code{mp\_bits\_per\_limb}}, b = 2^mp_bits_per_limb}, this remainder
@math{r} is of the form
@tex
$$ a = qd + r b^n $$
@end tex
@ifnottex

@example
a = q*d + r*b^n
@end example

@end ifnottex
@math{n} represents the number of zero limbs produced by the subtractions,
that being the number of limbs produced for @math{q}.  @math{r} will be in the
range @math{0@le{}r<d} and can be viewed as a remainder, but one shifted up by
a factor of @math{b^n}.

Carrying out full subtractions at each stage means the same number of cross
products must be done as a normal division, but there's still some single limb
divisions saved.  When @math{d} is a single limb some simplifications arise,
providing good speedups on a number of processors.

The functions @code{mpn_divexact_by3}, @code{mpn_modexact_1_odd} and the
internal @code{mpn_redc_X} functions differ subtly in how they return @math{r},
leading to some negations in the above formula, but all are essentially the
same.

@cindex Divisibility algorithm
@cindex Congruence algorithm
Clearly @math{r} is zero when @math{a} is a multiple of @math{d}, and this
leads to divisibility or congruence tests which are potentially more efficient
than a normal division.

The factor of @math{b^n} on @math{r} can be ignored in a GCD when @math{d} is
odd, hence the use of @code{mpn_modexact_1_odd} by @code{mpn_gcd_1} and
@code{mpz_kronecker_ui} etc (@pxref{Greatest Common Divisor Algorithms}).

Montgomery's REDC method for modular multiplications uses operands of the form
of @m{xb^{-n}, x*b^-n} and @m{yb^{-n}, y*b^-n} and on calculating @m{(xb^{-n})
(yb^{-n}), (x*b^-n)*(y*b^-n)} uses the factor of @math{b^n} in the exact
remainder to reach a product in the same form @m{(xy)b^{-n}, (x*y)*b^-n}
(@pxref{Modular Powering Algorithm}).

Notice that @math{r} generally gives no useful information about the ordinary
remainder @math{a @bmod d} since @math{b^n @bmod d} could be anything.  If
however @math{b^n @equiv{} 1 @bmod d}, then @math{r} is the negative of the
ordinary remainder.  This occurs whenever @math{d} is a factor of
@math{b^n-1}, as for example with 3 in @code{mpn_divexact_by3}.  For a 32 or
64 bit limb other such factors include 5, 17 and 257, but no particular use
has been found for this.


@node Small Quotient Division,  , Exact Remainder, Division Algorithms
@subsection Small Quotient Division

An N@cross{}M division where the number of quotient limbs Q=N@minus{}M is
small can be optimized somewhat.

An ordinary basecase division normalizes the divisor by shifting it to make
the high bit set, shifting the dividend accordingly, and shifting the
remainder back down at the end of the calculation.  This is wasteful if only a
few quotient limbs are to be formed.  Instead a division of just the top
@m{\rm2Q,2*Q} limbs of the dividend by the top Q limbs of the divisor can be
used to form a trial quotient.  This requires only those limbs normalized, not
the whole of the divisor and dividend.

A multiply and subtract then applies the trial quotient to the M@minus{}Q
unused limbs of the divisor and N@minus{}Q dividend limbs (which includes Q
limbs remaining from the trial quotient division).  The starting trial
quotient can be 1 or 2 too big, but all cases of 2 too big and most cases of 1
too big are detected by first comparing the most significant limbs that will
arise from the subtraction.  An addback is done if the quotient still turns
out to be 1 too big.

This whole procedure is essentially the same as one step of the basecase
algorithm done in a Q limb base, though with the trial quotient test done only
with the high limbs, not an entire Q limb ``digit'' product.  The correctness
of this weaker test can be established by following the argument of Knuth
section 4.3.1 exercise 20 but with the @m{v_2 \GMPhat q > b \GMPhat r
+ u_2, v2*q>b*r+u2} condition appropriately relaxed.


@need 1000
@node Greatest Common Divisor Algorithms, Powering Algorithms, Division Algorithms, Algorithms
@section Greatest Common Divisor
@cindex Greatest common divisor algorithms
@cindex GCD algorithms

@menu
* Binary GCD::
* Lehmer's Algorithm::
* Subquadratic GCD::
* Extended GCD::
* Jacobi Symbol::
@end menu


@node Binary GCD, Lehmer's Algorithm, Greatest Common Divisor Algorithms, Greatest Common Divisor Algorithms
@subsection Binary GCD

At small sizes GMP uses an @math{O(N^2)} binary style GCD@.  This is described
in many textbooks, for example Knuth section 4.5.2 algorithm B@.  It simply
consists of successively reducing odd operands @math{a} and @math{b} using

@quotation
@math{a,b = @abs{}(a-b),@min{}(a,b)} @*
strip factors of 2 from @math{a}
@end quotation

The Euclidean GCD algorithm, as per Knuth algorithms E and A, repeatedly
computes the quotient @m{q = \lfloor a/b \rfloor, q = floor(a/b)} and replaces
@math{a,b} by @math{v, u - q v}. The binary algorithm has so far been found to
be faster than the Euclidean algorithm everywhere.  One reason the binary
method does well is that the implied quotient at each step is usually small,
so often only one or two subtractions are needed to get the same effect as a
division.  Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth
section 4.5.3 Theorem E.

When the implied quotient is large, meaning @math{b} is much smaller than
@math{a}, then a division is worthwhile.  This is the basis for the initial
@math{a @bmod b} reductions in @code{mpn_gcd} and @code{mpn_gcd_1} (the latter
for both N@cross{}1 and 1@cross{}1 cases).  But after that initial reduction,
big quotients occur too rarely to make it worth checking for them.

@sp 1
The final @math{1@cross{}1} GCD in @code{mpn_gcd_1} is done in the generic C
code as described above.  For two N-bit operands, the algorithm takes about
0.68 iterations per bit.  For optimum performance some attention needs to be
paid to the way the factors of 2 are stripped from @math{a}.

Firstly it may be noted that in twos complement the number of low zero bits on
@math{a-b} is the same as @math{b-a}, so counting or testing can begin on
@math{a-b} without waiting for @math{@abs{}(a-b)} to be determined.

A loop stripping low zero bits tends not to branch predict well, since the
condition is data dependent.  But on average there's only a few low zeros, so
an option is to strip one or two bits arithmetically then loop for more (as
done for AMD K6).  Or use a lookup table to get a count for several bits then
loop for more (as done for AMD K7).  An alternative approach is to keep just
one of @math{a} or @math{b} odd and iterate

@quotation
@math{a,b = @abs{}(a-b), @min{}(a,b)} @*
@math{a = a/2} if even @*
@math{b = b/2} if even
@end quotation

This requires about 1.25 iterations per bit, but stripping of a single bit at
each step avoids any branching.  Repeating the bit strip reduces to about 0.9
iterations per bit, which may be a worthwhile tradeoff.

Generally with the above approaches a speed of perhaps 6 cycles per bit can be
achieved, which is still not terribly fast with for instance a 64-bit GCD
taking nearly 400 cycles.  It's this sort of time which means it's not usually
advantageous to combine a set of divisibility tests into a GCD.

Currently, the binary algorithm is used for GCD only when @math{N < 3}.

@node Lehmer's Algorithm, Subquadratic GCD, Binary GCD, Greatest Common Divisor Algorithms
@comment  node-name,  next,  previous,  up
@subsection Lehmer's algorithm

Lehmer's improvement of the Euclidean algorithms is based on the observation
that the initial part of the quotient sequence depends only on the most
significant parts of the inputs. The variant of Lehmer's algorithm used in GMP
splits off the most significant two limbs, as suggested, e.g., in ``A
Double-Digit Lehmer-Euclid Algorithm'' by Jebelean (@pxref{References}). The
quotients of two double-limb inputs are collected as a 2 by 2 matrix with
single-limb elements. This is done by the function @code{mpn_hgcd2}. The
resulting matrix is applied to the inputs using @code{mpn_mul_1} and
@code{mpn_submul_1}. Each iteration usually reduces the inputs by almost one
limb. In the rare case of a large quotient, no progress can be made by
examining just the most significant two limbs, and the quotient is computed
using plain division.

The resulting algorithm is asymptotically @math{O(N^2)}, just as the Euclidean
algorithm and the binary algorithm. The quadratic part of the work are
the calls to @code{mpn_mul_1} and @code{mpn_submul_1}. For small sizes, the
linear work is also significant. There are roughly @math{N} calls to the
@code{mpn_hgcd2} function. This function uses a couple of important
optimizations:

@itemize
@item
It uses the same relaxed notion of correctness as @code{mpn_hgcd} (see next
section). This means that when called with the most significant two limbs of
two large numbers, the returned matrix does not always correspond exactly to
the initial quotient sequence for the two large numbers; the final quotient
may sometimes be one off.

@item
It takes advantage of the fact the quotients are usually small. The division
operator is not used, since the corresponding assembler instruction is very
slow on most architectures. (This code could probably be improved further, it
uses many branches that are unfriendly to prediction).

@item
It switches from double-limb calculations to single-limb calculations half-way
through, when the input numbers have been reduced in size from two limbs to
one and a half.

@end itemize

@node Subquadratic GCD, Extended GCD, Lehmer's Algorithm, Greatest Common Divisor Algorithms
@subsection Subquadratic GCD

For inputs larger than @code{GCD_DC_THRESHOLD}, GCD is computed via the HGCD
(Half GCD) function, as a generalization to Lehmer's algorithm.

Let the inputs @math{a,b} be of size @math{N} limbs each. Put @m{S=\lfloor N/2
\rfloor + 1, S = floor(N/2) + 1}. Then HGCD(a,b) returns a transformation
matrix @math{T} with non-negative elements, and reduced numbers @math{(c;d) =
T^{-1} (a;b)}. The reduced numbers @math{c,d} must be larger than @math{S}
limbs, while their difference @math{abs(c-d)} must fit in @math{S} limbs. The
matrix elements will also be of size roughly @math{N/2}.

The HGCD base case uses Lehmer's algorithm, but with the above stop condition
that returns reduced numbers and the corresponding transformation matrix
half-way through. For inputs larger than @code{HGCD_THRESHOLD}, HGCD is
computed recursively, using the divide and conquer algorithm in ``On
Sch@"onhage's algorithm and subquadratic integer GCD computation'' by M@"oller
(@pxref{References}). The recursive algorithm consists of these main
steps.

@itemize

@item
Call HGCD recursively, on the most significant @math{N/2} limbs. Apply the
resulting matrix @math{T_1} to the full numbers, reducing them to a size just
above @math{3N/2}.

@item
Perform a small number of division or subtraction steps to reduce the numbers
to size below @math{3N/2}. This is essential mainly for the unlikely case of
large quotients.

@item
Call HGCD recursively, on the most significant @math{N/2} limbs of the reduced
numbers. Apply the resulting matrix @math{T_2} to the full numbers, reducing
them to a size just above @math{N/2}.

@item
Compute @math{T = T_1 T_2}.

@item
Perform a small number of division and subtraction steps to satisfy the
requirements, and return.
@end itemize

GCD is then implemented as a loop around HGCD, similarly to Lehmer's
algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
@code{mpn_hgcd2}, and applies the resulting matrix to the full numbers, the
sub-quadratic GCD chops off the most significant third of the limbs (the
proportion is a tuning parameter, and @math{1/3} seems to be more efficient
than, e.g, @math{1/2}), calls @code{mpn_hgcd}, and applies the resulting
matrix. Once the input numbers are reduced to size below
@code{GCD_DC_THRESHOLD}, Lehmer's algorithm is used for the rest of the work.

The asymptotic running time of both HGCD and GCD is @m{O(M(N)\log N),O(M(N)*log(N))},
where @math{M(N)} is the time for multiplying two @math{N}-limb numbers.

@comment  node-name,  next,  previous,  up

@node Extended GCD, Jacobi Symbol, Subquadratic GCD, Greatest Common Divisor Algorithms
@subsection Extended GCD

The extended GCD function, or GCDEXT, calculates @math{@gcd{}(a,b)} and also
cofactors @math{x} and @math{y} satisfying @m{ax+by=\gcd(a@C{}b),
a*x+b*y=gcd(a@C{}b)}. All the algorithms used for plain GCD are extended to
handle this case. The binary algorithm is used only for single-limb GCDEXT.
Lehmer's algorithm is used for sizes up to @code{GCDEXT_DC_THRESHOLD}. Above
this threshold, GCDEXT is implemented as a loop around HGCD, but with more
book-keeping to keep track of the cofactors. This gives the same asymptotic
running time as for GCD and HGCD, @m{O(M(N)\log N),O(M(N)*log(N))}

One difference to plain GCD is that while the inputs @math{a} and @math{b} are
reduced as the algorithm proceeds, the cofactors @math{x} and @math{y} grow in
size. This makes the tuning of the chopping-point more difficult. The current
code chops off the most significant half of the inputs for the call to HGCD in
the first iteration, and the most significant two thirds for the remaining
calls. This strategy could surely be improved. Also the stop condition for the
loop, where Lehmer's algorithm is invoked once the inputs are reduced below
@code{GCDEXT_DC_THRESHOLD}, could maybe be improved by taking into account the
current size of the cofactors.

@node Jacobi Symbol,  , Extended GCD, Greatest Common Divisor Algorithms
@subsection Jacobi Symbol
@cindex Jacobi symbol algorithm

@c Editor Note: I don't see other people defining the inputs, it would be nice
@c here because the code uses (a/b) where other references use (n/k)

Jacobi symbol @m{\left(a \over b\right), (@var{a}/@var{b})}

Initially if either operand fits in a single limb, a reduction is done with
either @code{mpn_mod_1} or @code{mpn_modexact_1_odd}, followed by the binary
algorithm on a single limb.  The binary algorithm is well suited to a single limb,
and the whole calculation in this case is quite efficient.

For inputs larger than @code{GCD_DC_THRESHOLD}, @code{mpz_jacobi},
@code{mpz_legendre} and @code{mpz_kronecker} are computed via the HGCD (Half
GCD) function, as a generalization to Lehmer's algorithm.

Most GCD algorithms reduce @math{a} and @math{b} by repeatatily computing the
quotient @m{q = \lfloor a/b \rfloor, q = floor(a/b)} and iteratively replacing

@c Couldn't figure out macros with commas.
@tex
$$ a, b = b, a - q * b$$
@end tex
@ifnottex
@math{a, b = b, a - q * b}
@end ifnottex

Different algorithms use different methods for calculating q, but the core
algorithm is the same if we use @ref{Lehmer's Algorithm} or
@ref{Subquadratic GCD, HGCD}.

At each step it is possible to compute if the reduction inverts the Jacobi
symbol based on the two least significant bits of @var{a} and @var{b}.  For
more details see ``Efficient computation of the Jacobi symbol'' by
M@"oller (@pxref{References}).

A small set of bits is thus used to track state
@itemize
@item
current sign of result (1 bit)

@item
two least significant bits of @var{a} and @var{b} (4 bits)

@item
a pointer to which input is currently the denominator (1 bit)
@end itemize

In all the routines sign changes for the result are accumulated using fast bit
twiddling which avoids conditional jumps.

The final result is calculated after verifying the inputs are coprime (GCD = 1)
by raising @m{(-1)^e,(-1)^e}

Much of the HGCD code is shared directly with the HGCD implementations, such
as the 2x2 matrix calculation, @xref{Lehmer's Algorithm} basecase and
@code{GCD_DC_THRESHOLD}.

The asymptotic running time is @m{O(M(N)\log N),O(M(N)*log(N))}, where
@math{M(N)} is the time for multiplying two @math{N}-limb numbers.

@need 1000
@node Powering Algorithms, Root Extraction Algorithms, Greatest Common Divisor Algorithms, Algorithms
@section Powering Algorithms
@cindex Powering algorithms

@menu
* Normal Powering Algorithm::
* Modular Powering Algorithm::
@end menu


@node Normal Powering Algorithm, Modular Powering Algorithm, Powering Algorithms, Powering Algorithms
@subsection Normal Powering

Normal @code{mpz} or @code{mpf} powering uses a simple binary algorithm,
successively squaring and then multiplying by the base when a 1 bit is seen in
the exponent, as per Knuth section 4.6.3.  The ``left to right''
variant described there is used rather than algorithm A, since it's just as
easy and can be done with somewhat less temporary memory.


@node Modular Powering Algorithm,  , Normal Powering Algorithm, Powering Algorithms
@subsection Modular Powering

Modular powering is implemented using a @math{2^k}-ary sliding window
algorithm, as per ``Handbook of Applied Cryptography'' algorithm 14.85
(@pxref{References}).  @math{k} is chosen according to the size of the
exponent.  Larger exponents use larger values of @math{k}, the choice being
made to minimize the average number of multiplications that must supplement
the squaring.

The modular multiplies and squarings use either a simple division or the REDC
method by Montgomery (@pxref{References}).  REDC is a little faster,
essentially saving N single limb divisions in a fashion similar to an exact
remainder (@pxref{Exact Remainder}).


@node Root Extraction Algorithms, Radix Conversion Algorithms, Powering Algorithms, Algorithms
@section Root Extraction Algorithms
@cindex Root extraction algorithms

@menu
* Square Root Algorithm::
* Nth Root Algorithm::
* Perfect Square Algorithm::
* Perfect Power Algorithm::
@end menu


@node Square Root Algorithm, Nth Root Algorithm, Root Extraction Algorithms, Root Extraction Algorithms
@subsection Square Root
@cindex Square root algorithm
@cindex Karatsuba square root algorithm

Square roots are taken using the ``Karatsuba Square Root'' algorithm by Paul
Zimmermann (@pxref{References}).

An input @math{n} is split into four parts of @math{k} bits each, so with
@math{b=2^k} we have @m{n = a_3b^3 + a_2b^2 + a_1b + a_0, n = a3*b^3 + a2*b^2
+ a1*b + a0}.  Part @ms{a,3} must be ``normalized'' so that either the high or
second highest bit is set.  In GMP, @math{k} is kept on a limb boundary and
the input is left shifted (by an even number of bits) to normalize.

The square root of the high two parts is taken, by recursive application of
the algorithm (bottoming out in a one-limb Newton's method),
@tex
$$ s',r' = \mathop{\rm sqrtrem} \> (a_3b + a_2) $$
@end tex
@ifnottex

@example
s1,r1 = sqrtrem (a3*b + a2)
@end example

@end ifnottex
This is an approximation to the desired root and is extended by a division to
give @math{s},@math{r},
@tex
$$\eqalign{
q,u &= \mathop{\rm divrem} \> (r'b + a_1, 2s') \cr
s &= s'b + q \cr
r &= ub + a_0 - q^2
}$$
@end tex
@ifnottex

@example
q,u = divrem (r1*b + a1, 2*s1)
s = s1*b + q
r = u*b + a0 - q^2
@end example

@end ifnottex
The normalization requirement on @ms{a,3} means at this point @math{s} is
either correct or 1 too big.  @math{r} is negative in the latter case, so
@tex
$$\eqalign{
\mathop{\rm if} \; r &< 0 \; \mathop{\rm then} \cr
r &\leftarrow r + 2s - 1 \cr
s &\leftarrow s - 1
}$$
@end tex
@ifnottex

@example
if r < 0 then
  r = r + 2*s - 1
  s = s - 1
@end example

@end ifnottex
The algorithm is expressed in a divide and conquer form, but as noted in the
paper it can also be viewed as a discrete variant of Newton's method, or as a
variation on the schoolboy method (no longer taught) for square roots two
digits at a time.

If the remainder @math{r} is not required then usually only a few high limbs
of @math{r} and @math{u} need to be calculated to determine whether an
adjustment to @math{s} is required.  This optimization is not currently
implemented.

In the Karatsuba multiplication range this algorithm is @m{O({3\over2}
M(N/2)),O(1.5*M(N/2))}, where @math{M(n)} is the time to multiply two numbers
of @math{n} limbs.  In the FFT multiplication range this grows to a bound of
@m{O(6 M(N/2)),O(6*M(N/2))}.  In practice a factor of about 1.5 to 1.8 is
found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.

The algorithm does all its calculations in integers and the resulting
@code{mpn_sqrtrem} is used for both @code{mpz_sqrt} and @code{mpf_sqrt}.
The extended precision given by @code{mpf_sqrt_ui} is obtained by
padding with zero limbs.


@node Nth Root Algorithm, Perfect Square Algorithm, Square Root Algorithm, Root Extraction Algorithms
@subsection Nth Root
@cindex Root extraction algorithm
@cindex Nth root algorithm

Integer Nth roots are taken using Newton's method with the following
iteration, where @math{A} is the input and @math{n} is the root to be taken.
@tex
$$a_{i+1} = {1\over n} \left({A \over a_i^{n-1}} + (n-1)a_i \right)$$
@end tex
@ifnottex

@example
         1         A
a[i+1] = - * ( --------- + (n-1)*a[i] )
         n     a[i]^(n-1)
@end example

@end ifnottex
The initial approximation @m{a_1,a[1]} is generated bitwise by successively
powering a trial root with or without new 1 bits, aiming to be just above the
true root.  The iteration converges quadratically when started from a good
approximation.  When @math{n} is large more initial bits are needed to get
good convergence.  The current implementation is not particularly well
optimized.


@node Perfect Square Algorithm, Perfect Power Algorithm, Nth Root Algorithm, Root Extraction Algorithms
@subsection Perfect Square
@cindex Perfect square algorithm

A significant fraction of non-squares can be quickly identified by checking
whether the input is a quadratic residue modulo small integers.

@code{mpz_perfect_square_p} first tests the input mod 256, which means just
examining the low byte.  Only 44 different values occur for squares mod 256,
so 82.8% of inputs can be immediately identified as non-squares.

On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for a total
99.25% of inputs identified as non-squares.  On a 64-bit system 97 is tested
too, for a total 99.62%.

These moduli are chosen because they're factors of @math{2^@W{24}-1} (or
@math{2^@W{48}-1} for 64-bits), and such a remainder can be quickly taken just
using additions (see @code{mpn_mod_34lsub1}).

When nails are in use moduli are instead selected by the @file{gen-psqr.c}
program and applied with an @code{mpn_mod_1}.  The same @math{2^@W{24}-1} or
@math{2^@W{48}-1} could be done with nails using some extra bit shifts, but
this is not currently implemented.

In any case each modulus is applied to the @code{mpn_mod_34lsub1} or
@code{mpn_mod_1} remainder and a table lookup identifies non-squares.  By
using a ``modexact'' style calculation, and suitably permuted tables, just one
multiply each is required, see the code for details.  Moduli are also combined
to save operations, so long as the lookup tables don't become too big.
@file{gen-psqr.c} does all the pre-calculations.

A square root must still be taken for any value that passes these tests, to
verify it's really a square and not one of the small fraction of non-squares
that get through (i.e.@: a pseudo-square to all the tested bases).

Clearly more residue tests could be done, @code{mpz_perfect_square_p} only
uses a compact and efficient set.  Big inputs would probably benefit from more
residue testing, small inputs might be better off with less.  The assumed
distribution of squares versus non-squares in the input would affect such
considerations.


@node Perfect Power Algorithm,  , Perfect Square Algorithm, Root Extraction Algorithms
@subsection Perfect Power
@cindex Perfect power algorithm

Detecting perfect powers is required by some factorization algorithms.
Currently @code{mpz_perfect_power_p} is implemented using repeated Nth root
extractions, though naturally only prime roots need to be considered.
(@xref{Nth Root Algorithm}.)

If a prime divisor @math{p} with multiplicity @math{e} can be found, then only
roots which are divisors of @math{e} need to be considered, much reducing the
work necessary.  To this end divisibility by a set of small primes is checked.


@node Radix Conversion Algorithms, Other Algorithms, Root Extraction Algorithms, Algorithms
@section Radix Conversion
@cindex Radix conversion algorithms

Radix conversions are less important than other algorithms.  A program
dominated by conversions should probably use a different data representation.

@menu
* Binary to Radix::
* Radix to Binary::
@end menu


@node Binary to Radix, Radix to Binary, Radix Conversion Algorithms, Radix Conversion Algorithms
@subsection Binary to Radix

Conversions from binary to a power-of-2 radix use a simple and fast
@math{O(N)} bit extraction algorithm.

Conversions from binary to other radices use one of two algorithms.  Sizes
below @code{GET_STR_PRECOMPUTE_THRESHOLD} use a basic @math{O(N^2)} method.
Repeated divisions by @math{b^n} are made, where @math{b} is the radix and
@math{n} is the biggest power that fits in a limb.  But instead of simply
using the remainder @math{r} from such divisions, an extra divide step is done
to give a fractional limb representing @math{r/b^n}.  The digits of @math{r}
can then be extracted using multiplications by @math{b} rather than divisions.
Special case code is provided for decimal, allowing multiplications by 10 to
optimize to shifts and adds.

Above @code{GET_STR_PRECOMPUTE_THRESHOLD} a sub-quadratic algorithm is used.
For an input @math{t}, powers @m{b^{n2^i},b^(n*2^i)} of the radix are
calculated, until a power between @math{t} and @m{\sqrt{t},sqrt(t)} is
reached.  @math{t} is then divided by that largest power, giving a quotient
which is the digits above that power, and a remainder which is those below.
These two parts are in turn divided by the second highest power, and so on
recursively.  When a piece has been divided down to less than
@code{GET_STR_DC_THRESHOLD} limbs, the basecase algorithm described above is
used.

The advantage of this algorithm is that big divisions can make use of the
sub-quadratic divide and conquer division (@pxref{Divide and Conquer
Division}), and big divisions tend to have less overheads than lots of
separate single limb divisions anyway.  But in any case the cost of
calculating the powers @m{b^{n2^i},b^(n*2^i)} must first be overcome.

@code{GET_STR_PRECOMPUTE_THRESHOLD} and @code{GET_STR_DC_THRESHOLD} represent
the same basic thing, the point where it becomes worth doing a big division to
cut the input in half.  @code{GET_STR_PRECOMPUTE_THRESHOLD} includes the cost
of calculating the radix power required, whereas @code{GET_STR_DC_THRESHOLD}
assumes that's already available, which is the case when recursing.

Since the base case produces digits from least to most significant but they
want to be stored from most to least, it's necessary to calculate in advance
how many digits there will be, or at least be sure not to underestimate that.
For GMP the number of input bits is multiplied by @code{chars_per_bit_exactly}
from @code{mp_bases}, rounding up.  The result is either correct or one too
big.

Examining some of the high bits of the input could increase the chance of
getting the exact number of digits, but an exact result every time would not
be practical, since in general the difference between numbers 100@dots{} and
99@dots{} is only in the last few bits and the work to identify 99@dots{}
might well be almost as much as a full conversion.

The @math{r/b^n} scheme described above for using multiplications to bring out
digits might be useful for more than a single limb.  Some brief experiments
with it on the base case when recursing didn't give a noticeable improvement,
but perhaps that was only due to the implementation.  Something similar would
work for the sub-quadratic divisions too, though there would be the cost of
calculating a bigger radix power.

Another possible improvement for the sub-quadratic part would be to arrange
for radix powers that balanced the sizes of quotient and remainder produced,
i.e.@: the highest power would be an @m{b^{nk},b^(n*k)} approximately equal to
@m{\sqrt{t},sqrt(t)}, not restricted to a @math{2^i} factor.  That ought to
smooth out a graph of times against sizes, but may or may not be a net
speedup.


@node Radix to Binary,  , Binary to Radix, Radix Conversion Algorithms
@subsection Radix to Binary

@strong{This section needs to be rewritten, it currently describes the
algorithms used before GMP 4.3.}

Conversions from a power-of-2 radix into binary use a simple and fast
@math{O(N)} bitwise concatenation algorithm.

Conversions from other radices use one of two algorithms.  Sizes below
@code{SET_STR_PRECOMPUTE_THRESHOLD} use a basic @math{O(N^2)} method.  Groups
of @math{n} digits are converted to limbs, where @math{n} is the biggest
power of the base @math{b} which will fit in a limb, then those groups are
accumulated into the result by multiplying by @math{b^n} and adding.  This
saves multi-precision operations, as per Knuth section 4.4 part E
(@pxref{References}).  Some special case code is provided for decimal, giving
the compiler a chance to optimize multiplications by 10.

Above @code{SET_STR_PRECOMPUTE_THRESHOLD} a sub-quadratic algorithm is used.
First groups of @math{n} digits are converted into limbs.  Then adjacent
limbs are combined into limb pairs with @m{xb^n+y,x*b^n+y}, where @math{x}
and @math{y} are the limbs.  Adjacent limb pairs are combined into quads
similarly with @m{xb^{2n}+y,x*b^(2n)+y}.  This continues until a single block
remains, that being the result.

The advantage of this method is that the multiplications for each @math{x} are
big blocks, allowing Karatsuba and higher algorithms to be used.  But the cost
of calculating the powers @m{b^{n2^i},b^(n*2^i)} must be overcome.
@code{SET_STR_PRECOMPUTE_THRESHOLD} usually ends up quite big, around 5000 digits, and on
some processors much bigger still.

@code{SET_STR_PRECOMPUTE_THRESHOLD} is based on the input digits (and tuned
for decimal), though it might be better based on a limb count, so as to be
independent of the base.  But that sort of count isn't used by the base case
and so would need some sort of initial calculation or estimate.

The main reason @code{SET_STR_PRECOMPUTE_THRESHOLD} is so much bigger than the
corresponding @code{GET_STR_PRECOMPUTE_THRESHOLD} is that @code{mpn_mul_1} is
much faster than @code{mpn_divrem_1} (often by a factor of 5, or more).


@need 1000
@node Other Algorithms, Assembly Coding, Radix Conversion Algorithms, Algorithms
@section Other Algorithms

@menu
* Prime Testing Algorithm::
* Factorial Algorithm::
* Binomial Coefficients Algorithm::
* Fibonacci Numbers Algorithm::
* Lucas Numbers Algorithm::
* Random Number Algorithms::
@end menu


@node Prime Testing Algorithm, Factorial Algorithm, Other Algorithms, Other Algorithms
@subsection Prime Testing
@cindex Prime testing algorithms

The primality testing in @code{mpz_probab_prime_p} (@pxref{Number Theoretic
Functions}) first does some trial division by small factors and then uses the
Miller-Rabin probabilistic primality testing algorithm, as described in Knuth
section 4.5.4 algorithm P (@pxref{References}).

For an odd input @math{n}, and with @math{n = q@GMPmultiply{}2^k+1} where
@math{q} is odd, this algorithm selects a random base @math{x} and tests
whether @math{x^q @bmod{} n} is 1 or @math{-1}, or an @m{x^{q2^j} \bmod n,
x^(q*2^j) mod n} is @math{1}, for @math{1@le{}j@le{}k}.  If so then @math{n}
is probably prime, if not then @math{n} is definitely composite.

Any prime @math{n} will pass the test, but some composites do too.  Such
composites are known as strong pseudoprimes to base @math{x}.  No @math{n} is
a strong pseudoprime to more than @math{1/4} of all bases (see Knuth exercise
22), hence with @math{x} chosen at random there's no more than a @math{1/4}
chance a ``probable prime'' will in fact be composite.

In fact strong pseudoprimes are quite rare, making the test much more
powerful than this analysis would suggest, but @math{1/4} is all that's proven
for an arbitrary @math{n}.


@node Factorial Algorithm, Binomial Coefficients Algorithm, Prime Testing Algorithm, Other Algorithms
@subsection Factorial
@cindex Factorial algorithm

Factorials are calculated by a combination of two algorithms. An idea is
shared among them: to compute the odd part of the factorial; a final step
takes account of the power of @math{2} term, by shifting.

For small @math{n}, the odd factor of @math{n!} is computed with the simple
observation that it is equal to the product of all positive odd numbers
smaller than @math{n} times the odd factor of @m{\lfloor n/2\rfloor!, [n/2]!},
where @m{\lfloor x\rfloor, [x]} is the integer part of @math{x}, and so on
recursively. The procedure can be best illustrated with an example,

@quotation
@math{23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19}}
@end quotation

Current code collects all the factors in a single list, with a loop and no
recursion, and compute the product, with no special care for repeated chunks.

When @math{n} is larger, computation pass trough prime sieving. An helper
function is used, as suggested by Peter Luschny:
@tex
$$\mathop{\rm msf}(n) = {n!\over\lfloor n/2\rfloor!^2\cdot2^k} = \prod_{p=3}^{n}
p^{\mathop{\rm L}(p,n)} $$
@end tex
@ifnottex

@example
                            n
                          -----
               n!          | |   L(p,n)
msf(n) = -------------- =  | |  p
          [n/2]!^2.2^k     p=3
@end example
@end ifnottex

Where @math{p} ranges on odd prime numbers. The exponent @math{k} is chosen to
obtain an odd integer number: @math{k} is the number of 1 bits in the binary
representation of @m{\lfloor n/2\rfloor, [n/2]}. The function L@math{(p,n)}
can be defined as zero when @math{p} is composite, and, for any prime
@math{p}, it is computed with:
@tex
$$\mathop{\rm L}(p,n) = \sum_{i>0}\left\lfloor{n\over p^i}\right\rfloor\bmod2
\leq\log_p(n)$$
@end tex
@ifnottex

@example
          ---
           \    n
L(p,n) =   /  [---] mod 2   <=  log (n) .
          ---  p^i                p
          i>0
@end example
@end ifnottex

With this helper function, we are able to compute the odd part of @math{n!}
using the recursion implied by @m{n!=\lfloor n/2\rfloor!^2\cdot\mathop{\rm
msf}(n)\cdot2^k , n!=[n/2]!^2*msf(n)*2^k}. The recursion stops using the
small-@math{n} algorithm on some @m{\lfloor n/2^i\rfloor, [n/2^i]}.

Both the above algorithms use binary splitting to compute the product of many
small factors. At first as many products as possible are accumulated in a
single register, generating a list of factors that fit in a machine word. This
list is then split into halves, and the product is computed recursively.

Such splitting is more efficient than repeated N@cross{}1 multiplies since it
forms big multiplies, allowing Karatsuba and higher algorithms to be used.
And even below the Karatsuba threshold a big block of work can be more
efficient for the basecase algorithm.


@node Binomial Coefficients Algorithm, Fibonacci Numbers Algorithm, Factorial Algorithm, Other Algorithms
@subsection Binomial Coefficients
@cindex Binomial coefficient algorithm

Binomial coefficients @m{\left({n}\atop{k}\right), C(n@C{}k)} are calculated
by first arranging @math{k @le{} n/2} using @m{\left({n}\atop{k}\right) =
\left({n}\atop{n-k}\right), C(n@C{}k) = C(n@C{}n-k)} if necessary, and then
evaluating the following product simply from @math{i=2} to @math{i=k}.
@tex
$$ \left({n}\atop{k}\right) = (n-k+1) \prod_{i=2}^{k} {{n-k+i} \over i} $$
@end tex
@ifnottex

@example
                      k  (n-k+i)
C(n,k) =  (n-k+1) * prod -------
                     i=2    i
@end example

@end ifnottex
It's easy to show that each denominator @math{i} will divide the product so
far, so the exact division algorithm is used (@pxref{Exact Division}).

The numerators @math{n-k+i} and denominators @math{i} are first accumulated
into as many fit a limb, to save multi-precision operations, though for
@code{mpz_bin_ui} this applies only to the divisors, since @math{n} is an
@code{mpz_t} and @math{n-k+i} in general won't fit in a limb at all.


@node Fibonacci Numbers Algorithm, Lucas Numbers Algorithm, Binomial Coefficients Algorithm, Other Algorithms
@subsection Fibonacci Numbers
@cindex Fibonacci number algorithm

The Fibonacci functions @code{mpz_fib_ui} and @code{mpz_fib2_ui} are designed
for calculating isolated @m{F_n,F[n]} or @m{F_n,F[n]},@m{F_{n-1},F[n-1]}
values efficiently.

For small @math{n}, a table of single limb values in @code{__gmp_fib_table} is
used.  On a 32-bit limb this goes up to @m{F_{47},F[47]}, or on a 64-bit limb
up to @m{F_{93},F[93]}.  For convenience the table starts at @m{F_{-1},F[-1]}.

Beyond the table, values are generated with a binary powering algorithm,
calculating a pair @m{F_n,F[n]} and @m{F_{n-1},F[n-1]} working from high to
low across the bits of @math{n}.  The formulas used are
@tex
$$\eqalign{
  F_{2k+1} &= 4F_k^2 - F_{k-1}^2 + 2(-1)^k \cr
  F_{2k-1} &=  F_k^2 + F_{k-1}^2           \cr
  F_{2k}   &= F_{2k+1} - F_{2k-1}
}$$
@end tex
@ifnottex

@example
F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
F[2k-1] =   F[k]^2 + F[k-1]^2

F[2k] = F[2k+1] - F[2k-1]
@end example

@end ifnottex
At each step, @math{k} is the high @math{b} bits of @math{n}.  If the next bit
of @math{n} is 0 then @m{F_{2k},F[2k]},@m{F_{2k-1},F[2k-1]} is used, or if
it's a 1 then @m{F_{2k+1},F[2k+1]},@m{F_{2k},F[2k]} is used, and the process
repeated until all bits of @math{n} are incorporated.  Notice these formulas
require just two squares per bit of @math{n}.

It'd be possible to handle the first few @math{n} above the single limb table
with simple additions, using the defining Fibonacci recurrence @m{F_{k+1} =
F_k + F_{k-1}, F[k+1]=F[k]+F[k-1]}, but this is not done since it usually
turns out to be faster for only about 10 or 20 values of @math{n}, and
including a block of code for just those doesn't seem worthwhile.  If they
really mattered it'd be better to extend the data table.

Using a table avoids lots of calculations on small numbers, and makes small
@math{n} go fast.  A bigger table would make more small @math{n} go fast, it's
just a question of balancing size against desired speed.  For GMP the code is
kept compact, with the emphasis primarily on a good powering algorithm.

@code{mpz_fib2_ui} returns both @m{F_n,F[n]} and @m{F_{n-1},F[n-1]}, but
@code{mpz_fib_ui} is only interested in @m{F_n,F[n]}.  In this case the last
step of the algorithm can become one multiply instead of two squares.  One of
the following two formulas is used, according as @math{n} is odd or even.
@tex
$$\eqalign{
  F_{2k}   &= F_k (F_k + 2F_{k-1}) \cr
  F_{2k+1} &= (2F_k + F_{k-1}) (2F_k - F_{k-1}) + 2(-1)^k
}$$
@end tex
@ifnottex

@example
F[2k]   = F[k]*(F[k]+2F[k-1])

F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
@end example

@end ifnottex
@m{F_{2k+1},F[2k+1]} here is the same as above, just rearranged to be a
multiply.  For interest, the @m{2(-1)^k, 2*(-1)^k} term both here and above
can be applied just to the low limb of the calculation, without a carry or
borrow into further limbs, which saves some code size.  See comments with
@code{mpz_fib_ui} and the internal @code{mpn_fib2_ui} for how this is done.


@node Lucas Numbers Algorithm, Random Number Algorithms, Fibonacci Numbers Algorithm, Other Algorithms
@subsection Lucas Numbers
@cindex Lucas number algorithm

@code{mpz_lucnum2_ui} derives a pair of Lucas numbers from a pair of Fibonacci
numbers with the following simple formulas.
@tex
$$\eqalign{
  L_k     &=  F_k + 2F_{k-1} \cr
  L_{k-1} &= 2F_k -  F_{k-1}
}$$
@end tex
@ifnottex

@example
L[k]   =   F[k] + 2*F[k-1]
L[k-1] = 2*F[k] -   F[k-1]
@end example

@end ifnottex
@code{mpz_lucnum_ui} is only interested in @m{L_n,L[n]}, and some work can be
saved.  Trailing zero bits on @math{n} can be handled with a single square
each.
@tex
$$ L_{2k} = L_k^2 - 2(-1)^k $$
@end tex
@ifnottex

@example
L[2k] = L[k]^2 - 2*(-1)^k
@end example

@end ifnottex
And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci
numbers, similar to what @code{mpz_fib_ui} does.
@tex
$$ L_{2k+1} = 5F_{k-1} (2F_k + F_{k-1}) - 4(-1)^k $$
@end tex
@ifnottex

@example
L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
@end example

@end ifnottex


@node Random Number Algorithms,  , Lucas Numbers Algorithm, Other Algorithms
@subsection Random Numbers
@cindex Random number algorithms

For the @code{urandomb} functions, random numbers are generated simply by
concatenating bits produced by the generator.  As long as the generator has
good randomness properties this will produce well-distributed @math{N} bit
numbers.

For the @code{urandomm} functions, random numbers in a range @math{0@le{}R<N}
are generated by taking values @math{R} of @m{\lceil \log_2 N \rceil,
ceil(log2(N))} bits each until one satisfies @math{R<N}.  This will normally
require only one or two attempts, but the attempts are limited in case the
generator is somehow degenerate and produces only 1 bits or similar.

@cindex Mersenne twister algorithm
The Mersenne Twister generator is by Matsumoto and Nishimura
(@pxref{References}).  It has a non-repeating period of @math{2^@W{19937}-1},
which is a Mersenne prime, hence the name of the generator.  The state is 624
words of 32-bits each, which is iterated with one XOR and shift for each
32-bit word generated, making the algorithm very fast.  Randomness properties
are also very good and this is the default algorithm used by GMP.

@cindex Linear congruential algorithm
Linear congruential generators are described in many text books, for instance
Knuth volume 2 (@pxref{References}).  With a modulus @math{M} and parameters
@math{A} and @math{C}, an integer state @math{S} is iterated by the formula
@math{S @leftarrow{} A@GMPmultiply{}S+C @bmod{} M}.  At each step the new
state is a linear function of the previous, mod @math{M}, hence the name of
the generator.

In GMP only moduli of the form @math{2^N} are supported, and the current
implementation is not as well optimized as it could be.  Overheads are
significant when @math{N} is small, and when @math{N} is large clearly the
multiply at each step will become slow.  This is not a big concern, since the
Mersenne Twister generator is better in every respect and is therefore
recommended for all normal applications.

For both generators the current state can be deduced by observing enough
output and applying some linear algebra (over GF(2) in the case of the
Mersenne Twister).  This generally means raw output is unsuitable for
cryptographic applications without further hashing or the like.


@node Assembly Coding,  , Other Algorithms, Algorithms
@section Assembly Coding
@cindex Assembly coding

The assembly subroutines in GMP are the most significant source of speed at
small to moderate sizes.  At larger sizes algorithm selection becomes more
important, but of course speedups in low level routines will still speed up
everything proportionally.

Carry handling and widening multiplies that are important for GMP can't be
easily expressed in C@.  GCC @code{asm} blocks help a lot and are provided in
@file{longlong.h}, but hand coding low level routines invariably offers a
speedup over generic C by a factor of anything from 2 to 10.

@menu
* Assembly Code Organisation::
* Assembly Basics::
* Assembly Carry Propagation::
* Assembly Cache Handling::
* Assembly Functional Units::
* Assembly Floating Point::
* Assembly SIMD Instructions::
* Assembly Software Pipelining::
* Assembly Loop Unrolling::
* Assembly Writing Guide::
@end menu


@node Assembly Code Organisation, Assembly Basics, Assembly Coding, Assembly Coding
@subsection Code Organisation
@cindex Assembly code organisation
@cindex Code organisation

The various @file{mpn} subdirectories contain machine-dependent code, written
in C or assembly.  The @file{mpn/generic} subdirectory contains default code,
used when there's no machine-specific version of a particular file.

Each @file{mpn} subdirectory is for an ISA family.  Generally 32-bit and
64-bit variants in a family cannot share code and have separate directories.
Within a family further subdirectories may exist for CPU variants.

In each directory a @file{nails} subdirectory may exist, holding code with
nails support for that CPU variant.  A @code{NAILS_SUPPORT} directive in each
file indicates the nails values the code handles.  Nails code only exists
where it's faster, or promises to be faster, than plain code.  There's no
effort put into nails if they're not going to enhance a given CPU.


@node Assembly Basics, Assembly Carry Propagation, Assembly Code Organisation, Assembly Coding
@subsection Assembly Basics

@code{mpn_addmul_1} and @code{mpn_submul_1} are the most important routines
for overall GMP performance.  All multiplications and divisions come down to
repeated calls to these.  @code{mpn_add_n}, @code{mpn_sub_n},
@code{mpn_lshift} and @code{mpn_rshift} are next most important.

On some CPUs assembly versions of the internal functions
@code{mpn_mul_basecase} and @code{mpn_sqr_basecase} give significant speedups,
mainly through avoiding function call overheads.  They can also potentially
make better use of a wide superscalar processor, as can bigger primitives like
@code{mpn_addmul_2} or @code{mpn_addmul_4}.

The restrictions on overlaps between sources and destinations
(@pxref{Low-level Functions}) are designed to facilitate a variety of
implementations.  For example, knowing @code{mpn_add_n} won't have partly
overlapping sources and destination means reading can be done far ahead of
writing on superscalar processors, and loops can be vectorized on a vector
processor, depending on the carry handling.


@node Assembly Carry Propagation, Assembly Cache Handling, Assembly Basics, Assembly Coding
@subsection Carry Propagation
@cindex Assembly carry propagation

The problem that presents most challenges in GMP is propagating carries from
one limb to the next.  In functions like @code{mpn_addmul_1} and
@code{mpn_add_n}, carries are the only dependencies between limb operations.

On processors with carry flags, a straightforward CISC style @code{adc} is
generally best.  AMD K6 @code{mpn_addmul_1} however is an example of an
unusual set of circumstances where a branch works out better.

On RISC processors generally an add and compare for overflow is used.  This
sort of thing can be seen in @file{mpn/generic/aors_n.c}.  Some carry
propagation schemes require 4 instructions, meaning at least 4 cycles per
limb, but other schemes may use just 1 or 2.  On wide superscalar processors
performance may be completely determined by the number of dependent
instructions between carry-in and carry-out for each limb.

On vector processors good use can be made of the fact that a carry bit only
very rarely propagates more than one limb.  When adding a single bit to a
limb, there's only a carry out if that limb was @code{0xFF@dots{}FF} which on
random data will be only 1 in @m{2\GMPraise{@code{mp\_bits\_per\_limb}},
2^mp_bits_per_limb}.  @file{mpn/cray/add_n.c} is an example of this, it adds
all limbs in parallel, adds one set of carry bits in parallel and then only
rarely needs to fall through to a loop propagating further carries.

On the x86s, GCC (as of version 2.95.2) doesn't generate particularly good code
for the RISC style idioms that are necessary to handle carry bits in
C@.  Often conditional jumps are generated where @code{adc} or @code{sbb} forms
would be better.  And so unfortunately almost any loop involving carry bits
needs to be coded in assembly for best results.


@node Assembly Cache Handling, Assembly Functional Units, Assembly Carry Propagation, Assembly Coding
@subsection Cache Handling
@cindex Assembly cache handling

GMP aims to perform well both on operands that fit entirely in L1 cache and
those which don't.

Basic routines like @code{mpn_add_n} or @code{mpn_lshift} are often used on
large operands, so L2 and main memory performance is important for them.
@code{mpn_mul_1} and @code{mpn_addmul_1} are mostly used for multiply and
square basecases, so L1 performance matters most for them, unless assembly
versions of @code{mpn_mul_basecase} and @code{mpn_sqr_basecase} exist, in
which case the remaining uses are mostly for larger operands.

For L2 or main memory operands, memory access times will almost certainly be
more than the calculation time.  The aim therefore is to maximize memory
throughput, by starting a load of the next cache line while processing the
contents of the previous one.  Clearly this is only possible if the chip has a
lock-up free cache or some sort of prefetch instruction.  Most current chips
have both these features.

Prefetching sources combines well with loop unrolling, since a prefetch can be
initiated once per unrolled loop (or more than once if the loop covers more
than one cache line).

On CPUs without write-allocate caches, prefetching destinations will ensure
individual stores don't go further down the cache hierarchy, limiting
bandwidth.  Of course for calculations which are slow anyway, like
@code{mpn_divrem_1}, write-throughs might be fine.

The distance ahead to prefetch will be determined by memory latency versus
throughput.  The aim of course is to have data arriving continuously, at peak
throughput.  Some CPUs have limits on the number of fetches or prefetches in
progress.

If a special prefetch instruction doesn't exist then a plain load can be used,
but in that case care must be taken not to attempt to read past the end of an
operand, since that might produce a segmentation violation.

Some CPUs or systems have hardware that detects sequential memory accesses and
initiates suitable cache movements automatically, making life easy.


@node Assembly Functional Units, Assembly Floating Point, Assembly Cache Handling, Assembly Coding
@subsection Functional Units

When choosing an approach for an assembly loop, consideration is given to
what operations can execute simultaneously and what throughput can thereby be
achieved.  In some cases an algorithm can be tweaked to accommodate available
resources.

Loop control will generally require a counter and pointer updates, costing as
much as 5 instructions, plus any delays a branch introduces.  CPU addressing
modes might reduce pointer updates, perhaps by allowing just one updating
pointer and others expressed as offsets from it, or on CISC chips with all
addressing done with the loop counter as a scaled index.

The final loop control cost can be amortised by processing several limbs in
each iteration (@pxref{Assembly Loop Unrolling}).  This at least ensures loop
control isn't a big fraction the work done.

Memory throughput is always a limit.  If perhaps only one load or one store
can be done per cycle then 3 cycles/limb will the top speed for ``binary''
operations like @code{mpn_add_n}, and any code achieving that is optimal.

Integer resources can be freed up by having the loop counter in a float
register, or by pressing the float units into use for some multiplying,
perhaps doing every second limb on the float side (@pxref{Assembly Floating
Point}).

Float resources can be freed up by doing carry propagation on the integer
side, or even by doing integer to float conversions in integers using bit
twiddling.


@node Assembly Floating Point, Assembly SIMD Instructions, Assembly Functional Units, Assembly Coding
@subsection Floating Point
@cindex Assembly floating Point

Floating point arithmetic is used in GMP for multiplications on CPUs with poor
integer multipliers.  It's mostly useful for @code{mpn_mul_1},
@code{mpn_addmul_1} and @code{mpn_submul_1} on 64-bit machines, and
@code{mpn_mul_basecase} on both 32-bit and 64-bit machines.

With IEEE 53-bit double precision floats, integer multiplications producing up
to 53 bits will give exact results.  Breaking a 64@cross{}64 multiplication
into eight 16@cross{}@math{32@rightarrow{}48} bit pieces is convenient.  With
some care though six 21@cross{}@math{32@rightarrow{}53} bit products can be
used, if one of the lower two 21-bit pieces also uses the sign bit.

For the @code{mpn_mul_1} family of functions on a 64-bit machine, the
invariant single limb is split at the start, into 3 or 4 pieces.  Inside the
loop, the bignum operand is split into 32-bit pieces.  Fast conversion of
these unsigned 32-bit pieces to floating point is highly machine-dependent.
In some cases, reading the data into the integer unit, zero-extending to
64-bits, then transferring to the floating point unit back via memory is the
only option.

Converting partial products back to 64-bit limbs is usually best done as a
signed conversion.  Since all values are smaller than @m{2^{53},2^53}, signed
and unsigned are the same, but most processors lack unsigned conversions.

@sp 2

Here is a diagram showing 16@cross{}32 bit products for an @code{mpn_mul_1} or
@code{mpn_addmul_1} with a 64-bit limb.  The single limb operand V is split
into four 16-bit parts.  The multi-limb operand U is split in the loop into
two 32-bit parts.

@tex
\global\newdimen\GMPbits      \global\GMPbits=0.18em
\def\GMPbox#1#2#3{%
  \hbox{%
    \hbox to 128\GMPbits{\hfil
      \vbox{%
        \hrule
        \hbox to 48\GMPbits {\GMPvrule \hfil$#2$\hfil \vrule}%
        \hrule}%
      \hskip #1\GMPbits}%
    \raise \GMPboxdepth \hbox{\hskip 2em #3}}}
%
\GMPdisplay{%
  \vbox{%
    \hbox{%
      \hbox to 128\GMPbits {\hfil
        \vbox{%
          \hrule
          \hbox to 64\GMPbits{%
            \GMPvrule \hfil$v48$\hfil
            \vrule    \hfil$v32$\hfil
            \vrule    \hfil$v16$\hfil
            \vrule    \hfil$v00$\hfil
            \vrule}
          \hrule}}%
       \raise \GMPboxdepth \hbox{\hskip 2em V Operand}}
    \vskip 0.5ex
    \hbox{%
      \hbox to 128\GMPbits {\hfil
        \raise \GMPboxdepth \hbox{$\times$\hskip 1.5em}%
        \vbox{%
          \hrule
          \hbox to 64\GMPbits {%
            \GMPvrule \hfil$u32$\hfil
            \vrule \hfil$u00$\hfil
            \vrule}%
          \hrule}}%
       \raise \GMPboxdepth \hbox{\hskip 2em U Operand (one limb)}}%
    \vskip 0.5ex
    \hbox{\vbox to 2ex{\hrule width 128\GMPbits}}%
    \GMPbox{0}{u00 \times v00}{$p00$\hskip 1.5em 48-bit products}%
    \vskip 0.5ex
    \GMPbox{16}{u00 \times v16}{$p16$}
    \vskip 0.5ex
    \GMPbox{32}{u00 \times v32}{$p32$}
    \vskip 0.5ex
    \GMPbox{48}{u00 \times v48}{$p48$}
    \vskip 0.5ex
    \GMPbox{32}{u32 \times v00}{$r32$}
    \vskip 0.5ex
    \GMPbox{48}{u32 \times v16}{$r48$}
    \vskip 0.5ex
    \GMPbox{64}{u32 \times v32}{$r64$}
    \vskip 0.5ex
    \GMPbox{80}{u32 \times v48}{$r80$}
}}
@end tex
@ifnottex
@example
@group
                +---+---+---+---+
                |v48|v32|v16|v00|    V operand
                +---+---+---+---+

                +-------+---+---+
            x   |  u32  |  u00  |    U operand (one limb)
                +---------------+

---------------------------------

                    +-----------+
                    | u00 x v00 |    p00    48-bit products
                    +-----------+
                +-----------+
                | u00 x v16 |        p16
                +-----------+
            +-----------+
            | u00 x v32 |            p32
            +-----------+
        +-----------+
        | u00 x v48 |                p48
        +-----------+
            +-----------+
            | u32 x v00 |            r32
            +-----------+
        +-----------+
        | u32 x v16 |                r48
        +-----------+
    +-----------+
    | u32 x v32 |                    r64
    +-----------+
+-----------+
| u32 x v48 |                        r80
+-----------+
@end group
@end example
@end ifnottex

@math{p32} and @math{r32} can be summed using floating-point addition, and
likewise @math{p48} and @math{r48}.  @math{p00} and @math{p16} can be summed
with @math{r64} and @math{r80} from the previous iteration.

For each loop then, four 49-bit quantities are transferred to the integer unit,
aligned as follows,

@tex
% GMPbox here should be 49 bits wide, but use 51 to better show p16+r80'
% crossing into the upper 64 bits.
\def\GMPbox#1#2#3{%
  \hbox{%
    \hbox to 128\GMPbits {%
      \hfil
      \vbox{%
        \hrule
        \hbox to 51\GMPbits {\GMPvrule \hfil$#2$\hfil \vrule}%
        \hrule}%
      \hskip #1\GMPbits}%
    \raise \GMPboxdepth \hbox{\hskip 1.5em $#3$\hfil}%
}}
\newbox\b \setbox\b\hbox{64 bits}%
\newdimen\bw \bw=\wd\b \advance\bw by 2em
\newdimen\x \x=128\GMPbits
\advance\x by -2\bw
\divide\x by4
\GMPdisplay{%
  \vbox{%
    \hbox to 128\GMPbits {%
      \GMPvrule
      \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
      \hfil 64 bits\hfil
      \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
      \vrule
      \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
      \hfil 64 bits\hfil
      \raise 0.5ex \vbox{\hrule \hbox to \x {}}%
      \vrule}%
    \vskip 0.7ex
    \GMPbox{0}{p00+r64'}{i00}
    \vskip 0.5ex
    \GMPbox{16}{p16+r80'}{i16}
    \vskip 0.5ex
    \GMPbox{32}{p32+r32}{i32}
    \vskip 0.5ex
    \GMPbox{48}{p48+r48}{i48}
}}
@end tex
@ifnottex
@example
@group
|-----64bits----|-----64bits----|
                   +------------+
                   | p00 + r64' |    i00
                   +------------+
               +------------+
               | p16 + r80' |        i16
               +------------+
           +------------+
           | p32 + r32  |            i32
           +------------+
       +------------+
       | p48 + r48  |                i48
       +------------+
@end group
@end example
@end ifnottex

The challenge then is to sum these efficiently and add in a carry limb,
generating a low 64-bit result limb and a high 33-bit carry limb (@math{i48}
extends 33 bits into the high half).


@node Assembly SIMD Instructions, Assembly Software Pipelining, Assembly Floating Point, Assembly Coding
@subsection SIMD Instructions
@cindex Assembly SIMD

The single-instruction multiple-data support in current microprocessors is
aimed at signal processing algorithms where each data point can be treated
more or less independently.  There's generally not much support for
propagating the sort of carries that arise in GMP.

SIMD multiplications of say four 16@cross{}16 bit multiplies only do as much
work as one 32@cross{}32 from GMP's point of view, and need some shifts and
adds besides.  But of course if say the SIMD form is fully pipelined and uses
less instruction decoding then it may still be worthwhile.

On the x86 chips, MMX has so far found a use in @code{mpn_rshift} and
@code{mpn_lshift}, and is used in a special case for 16-bit multipliers in the
P55 @code{mpn_mul_1}.  SSE2 is used for Pentium 4 @code{mpn_mul_1},
@code{mpn_addmul_1}, and @code{mpn_submul_1}.


@node Assembly Software Pipelining, Assembly Loop Unrolling, Assembly SIMD Instructions, Assembly Coding
@subsection Software Pipelining
@cindex Assembly software pipelining

Software pipelining consists of scheduling instructions around the branch
point in a loop.  For example a loop might issue a load not for use in the
present iteration but the next, thereby allowing extra cycles for the data to
arrive from memory.

Naturally this is wanted only when doing things like loads or multiplies that
take several cycles to complete, and only where a CPU has multiple functional
units so that other work can be done in the meantime.

A pipeline with several stages will have a data value in progress at each
stage and each loop iteration moves them along one stage.  This is like
juggling.

If the latency of some instruction is greater than the loop time then it will
be necessary to unroll, so one register has a result ready to use while
another (or multiple others) are still in progress.  (@pxref{Assembly Loop
Unrolling}).


@node Assembly Loop Unrolling, Assembly Writing Guide, Assembly Software Pipelining, Assembly Coding
@subsection Loop Unrolling
@cindex Assembly loop unrolling

Loop unrolling consists of replicating code so that several limbs are
processed in each loop.  At a minimum this reduces loop overheads by a
corresponding factor, but it can also allow better register usage, for example
alternately using one register combination and then another.  Judicious use of
@command{m4} macros can help avoid lots of duplication in the source code.

Any amount of unrolling can be handled with a loop counter that's decremented
by @math{N} each time, stopping when the remaining count is less than the
further @math{N} the loop will process.  Or by subtracting @math{N} at the
start, the termination condition becomes when the counter @math{C} is less
than 0 (and the count of remaining limbs is @math{C+N}).

Alternately for a power of 2 unroll the loop count and remainder can be
established with a shift and mask.  This is convenient if also making a
computed jump into the middle of a large loop.

The limbs not a multiple of the unrolling can be handled in various ways, for
example

@itemize @bullet
@item
A simple loop at the end (or the start) to process the excess.  Care will be
wanted that it isn't too much slower than the unrolled part.

@item
A set of binary tests, for example after an 8-limb unrolling, test for 4 more
limbs to process, then a further 2 more or not, and finally 1 more or not.
This will probably take more code space than a simple loop.

@item
A @code{switch} statement, providing separate code for each possible excess,
for example an 8-limb unrolling would have separate code for 0 remaining, 1
remaining, etc, up to 7 remaining.  This might take a lot of code, but may be
the best way to optimize all cases in combination with a deep pipelined loop.

@item
A computed jump into the middle of the loop, thus making the first iteration
handle the excess.  This should make times smoothly increase with size, which
is attractive, but setups for the jump and adjustments for pointers can be
tricky and could become quite difficult in combination with deep pipelining.
@end itemize


@node Assembly Writing Guide,  , Assembly Loop Unrolling, Assembly Coding
@subsection Writing Guide
@cindex Assembly writing guide

This is a guide to writing software pipelined loops for processing limb
vectors in assembly.

First determine the algorithm and which instructions are needed.  Code it
without unrolling or scheduling, to make sure it works.  On a 3-operand CPU
try to write each new value to a new register, this will greatly simplify later
steps.

Then note for each instruction the functional unit and/or issue port
requirements.  If an instruction can use either of two units, like U0 or U1
then make a category ``U0/U1''.  Count the total using each unit (or combined
unit), and count all instructions.

Figure out from those counts the best possible loop time.  The goal will be to
find a perfect schedule where instruction latencies are completely hidden.
The total instruction count might be the limiting factor, or perhaps a
particular functional unit.  It might be possible to tweak the instructions to
help the limiting factor.

Suppose the loop time is @math{N}, then make @math{N} issue buckets, with the
final loop branch at the end of the last.  Now fill the buckets with dummy
instructions using the functional units desired.  Run this to make sure the
intended speed is reached.

Now replace the dummy instructions with the real instructions from the slow
but correct loop you started with.  The first will typically be a load
instruction.  Then the instruction using that value is placed in a bucket an
appropriate distance down.  Run the loop again, to check it still runs at
target speed.

Keep placing instructions, frequently measuring the loop.  After a few you
will need to wrap around from the last bucket back to the top of the loop.  If
you used the new-register for new-value strategy above then there will be no
register conflicts.  If not then take care not to clobber something already in
use.  Changing registers at this time is very error prone.

The loop will overlap two or more of the original loop iterations, and the
computation of one vector element result will be started in one iteration of
the new loop, and completed one or several iterations later.

The final step is to create feed-in and wind-down code for the loop.  A good
way to do this is to make a copy (or copies) of the loop at the start and
delete those instructions which don't have valid antecedents, and at the end
replicate and delete those whose results are unwanted (including any further
loads).

The loop will have a minimum number of limbs loaded and processed, so the
feed-in code must test if the request size is smaller and skip either to a
suitable part of the wind-down or to special code for small sizes.


@node Internals, Contributors, Algorithms, Top
@chapter Internals
@cindex Internals

@strong{This chapter is provided only for informational purposes and the
various internals described here may change in future GMP releases.
Applications expecting to be compatible with future releases should use only
the documented interfaces described in previous chapters.}

@menu
* Integer Internals::
* Rational Internals::
* Float Internals::
* Raw Output Internals::
* C++ Interface Internals::
@end menu

@node Integer Internals, Rational Internals, Internals, Internals
@section Integer Internals
@cindex Integer internals

@code{mpz_t} variables represent integers using sign and magnitude, in space
dynamically allocated and reallocated.  The fields are as follows.

@table @asis
@item @code{_mp_size}
The number of limbs, or the negative of that when representing a negative
integer.  Zero is represented by @code{_mp_size} set to zero, in which case
the @code{_mp_d} data is undefined.

@item @code{_mp_d}
A pointer to an array of limbs which is the magnitude.  These are stored
``little endian'' as per the @code{mpn} functions, so @code{_mp_d[0]} is the
least significant limb and @code{_mp_d[ABS(_mp_size)-1]} is the most
significant.  Whenever @code{_mp_size} is non-zero, the most significant limb
is non-zero.

Currently there's always at least one readable limb, so for instance
@code{mpz_get_ui} can fetch @code{_mp_d[0]} unconditionally (though its value
is undefined if @code{_mp_size} is zero).

@item @code{_mp_alloc}
@code{_mp_alloc} is the number of limbs currently allocated at @code{_mp_d},
and normally @code{_mp_alloc >= ABS(_mp_size)}.  When an @code{mpz} routine
is about to (or might be about to) increase @code{_mp_size}, it checks
@code{_mp_alloc} to see whether there's enough space, and reallocates if not.
@code{MPZ_REALLOC} is generally used for this.

@code{mpz_t} variables initialised with the @code{mpz_roinit_n} function or
the @code{MPZ_ROINIT_N} macro have @code{_mp_alloc = 0} but can have a
non-zero @code{_mp_size}.  They can only be used as read-only constants. See
@ref{Integer Special Functions} for details.
@end table

The various bitwise logical functions like @code{mpz_and} behave as if
negative values were twos complement.  But sign and magnitude is always used
internally, and necessary adjustments are made during the calculations.
Sometimes this isn't pretty, but sign and magnitude are best for other
routines.

Some internal temporary variables are setup with @code{MPZ_TMP_INIT} and these
have @code{_mp_d} space obtained from @code{TMP_ALLOC} rather than the memory
allocation functions.  Care is taken to ensure that these are big enough that
no reallocation is necessary (since it would have unpredictable consequences).

@code{_mp_size} and @code{_mp_alloc} are @code{int}, although @code{mp_size_t}
is usually a @code{long}.  This is done to make the fields just 32 bits on
some 64 bits systems, thereby saving a few bytes of data space but still
providing plenty of range.


@node Rational Internals, Float Internals, Integer Internals, Internals
@section Rational Internals
@cindex Rational internals

@code{mpq_t} variables represent rationals using an @code{mpz_t} numerator and
denominator (@pxref{Integer Internals}).

The canonical form adopted is denominator positive (and non-zero), no common
factors between numerator and denominator, and zero uniquely represented as
0/1.

It's believed that casting out common factors at each stage of a calculation
is best in general.  A GCD is an @math{O(N^2)} operation so it's better to do
a few small ones immediately than to delay and have to do a big one later.
Knowing the numerator and denominator have no common factors can be used for
example in @code{mpq_mul} to make only two cross GCDs necessary, not four.

This general approach to common factors is badly sub-optimal in the presence
of simple factorizations or little prospect for cancellation, but GMP has no
way to know when this will occur.  As per @ref{Efficiency}, that's left to
applications.  The @code{mpq_t} framework might still suit, with
@code{mpq_numref} and @code{mpq_denref} for direct access to the numerator and
denominator, or of course @code{mpz_t} variables can be used directly.


@node Float Internals, Raw Output Internals, Rational Internals, Internals
@section Float Internals
@cindex Float internals

Efficient calculation is the primary aim of GMP floats and the use of whole
limbs and simple rounding facilitates this.

@code{mpf_t} floats have a variable precision mantissa and a single machine
word signed exponent.  The mantissa is represented using sign and magnitude.

@c FIXME: The arrow heads don't join to the lines exactly.
@tex
\global\newdimen\GMPboxwidth \GMPboxwidth=5em
\global\newdimen\GMPboxheight \GMPboxheight=3ex
\def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
\GMPdisplay{%
\vbox{%
  \hbox to 5\GMPboxwidth {most significant limb \hfil least significant limb}
  \vskip 0.7ex
  \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
  \hbox {
    \hbox to 3\GMPboxwidth {%
      \setbox 0 = \hbox{@code{\_mp\_exp}}%
      \dimen0=3\GMPboxwidth
      \advance\dimen0 by -\wd0
      \divide\dimen0 by 2
      \advance\dimen0 by -1em
      \setbox1 = \hbox{$\rightarrow$}%
      \dimen1=\dimen0
      \advance\dimen1 by -\wd1
      \GMPcentreline{\dimen0}%
      \hfil
      \box0%
      \hfil
      \GMPcentreline{\dimen1{}}%
      \box1}
    \hbox to 2\GMPboxwidth {\hfil @code{\_mp\_d}}}
  \vskip 0.5ex
  \vbox {%
    \hrule
    \hbox{%
      \vrule height 2ex depth 1ex
      \hbox to \GMPboxwidth {}%
      \vrule
      \hbox to \GMPboxwidth {}%
      \vrule
      \hbox to \GMPboxwidth {}%
      \vrule
      \hbox to \GMPboxwidth {}%
      \vrule
      \hbox to \GMPboxwidth {}%
      \vrule}
    \hrule
  }
  \hbox {%
    \hbox to 0.8 pt {}
    \hbox to 3\GMPboxwidth {%
      \hfil $\cdot$} \hbox {$\leftarrow$ radix point\hfil}}
  \hbox to 5\GMPboxwidth{%
    \setbox 0 = \hbox{@code{\_mp\_size}}%
    \dimen0 = 5\GMPboxwidth
    \advance\dimen0 by -\wd0
    \divide\dimen0 by 2
    \advance\dimen0 by -1em
    \dimen1 = \dimen0
    \setbox1 = \hbox{$\leftarrow$}%
    \setbox2 = \hbox{$\rightarrow$}%
    \advance\dimen0 by -\wd1
    \advance\dimen1 by -\wd2
    \hbox to 0.3 em {}%
    \box1
    \GMPcentreline{\dimen0}%
    \hfil
    \box0
    \hfil
    \GMPcentreline{\dimen1}%
    \box2}
}}
@end tex
@ifnottex
@example
   most                   least
significant            significant
   limb                   limb

                            _mp_d
 |---- _mp_exp --->           |
  _____ _____ _____ _____ _____
 |_____|_____|_____|_____|_____|
                   . <------------ radix point

  <-------- _mp_size --------->
@sp 1
@end example
@end ifnottex

@noindent
The fields are as follows.

@table @asis
@item @code{_mp_size}
The number of limbs currently in use, or the negative of that when
representing a negative value.  Zero is represented by @code{_mp_size} and
@code{_mp_exp} both set to zero, and in that case the @code{_mp_d} data is
unused.  (In the future @code{_mp_exp} might be undefined when representing
zero.)

@item @code{_mp_prec}
The precision of the mantissa, in limbs.  In any calculation the aim is to
produce @code{_mp_prec} limbs of result (the most significant being non-zero).

@item @code{_mp_d}
A pointer to the array of limbs which is the absolute value of the mantissa.
These are stored ``little endian'' as per the @code{mpn} functions, so
@code{_mp_d[0]} is the least significant limb and
@code{_mp_d[ABS(_mp_size)-1]} the most significant.

The most significant limb is always non-zero, but there are no other
restrictions on its value, in particular the highest 1 bit can be anywhere
within the limb.

@code{_mp_prec+1} limbs are allocated to @code{_mp_d}, the extra limb being
for convenience (see below).  There are no reallocations during a calculation,
only in a change of precision with @code{mpf_set_prec}.

@item @code{_mp_exp}
The exponent, in limbs, determining the location of the implied radix point.
Zero means the radix point is just above the most significant limb.  Positive
values mean a radix point offset towards the lower limbs and hence a value
@math{@ge{} 1}, as for example in the diagram above.  Negative exponents mean
a radix point further above the highest limb.

Naturally the exponent can be any value, it doesn't have to fall within the
limbs as the diagram shows, it can be a long way above or a long way below.
Limbs other than those included in the @code{@{_mp_d,_mp_size@}} data
are treated as zero.
@end table

The @code{_mp_size} and @code{_mp_prec} fields are @code{int}, although the
@code{mp_size_t} type is usually a @code{long}.  The @code{_mp_exp} field is
usually @code{long}.  This is done to make some fields just 32 bits on some 64
bits systems, thereby saving a few bytes of data space but still providing
plenty of precision and a very large range.


@sp 1
@noindent
The following various points should be noted.

@table @asis
@item Low Zeros
The least significant limbs @code{_mp_d[0]} etc can be zero, though such low
zeros can always be ignored.  Routines likely to produce low zeros check and
avoid them to save time in subsequent calculations, but for most routines
they're quite unlikely and aren't checked.

@item Mantissa Size Range
The @code{_mp_size} count of limbs in use can be less than @code{_mp_prec} if
the value can be represented in less.  This means low precision values or
small integers stored in a high precision @code{mpf_t} can still be operated
on efficiently.

@code{_mp_size} can also be greater than @code{_mp_prec}.  Firstly a value is
allowed to use all of the @code{_mp_prec+1} limbs available at @code{_mp_d},
and secondly when @code{mpf_set_prec_raw} lowers @code{_mp_prec} it leaves
@code{_mp_size} unchanged and so the size can be arbitrarily bigger than
@code{_mp_prec}.

@item Rounding
All rounding is done on limb boundaries.  Calculating @code{_mp_prec} limbs
with the high non-zero will ensure the application requested minimum precision
is obtained.

The use of simple ``trunc'' rounding towards zero is efficient, since there's
no need to examine extra limbs and increment or decrement.

@item Bit Shifts
Since the exponent is in limbs, there are no bit shifts in basic operations
like @code{mpf_add} and @code{mpf_mul}.  When differing exponents are
encountered all that's needed is to adjust pointers to line up the relevant
limbs.

Of course @code{mpf_mul_2exp} and @code{mpf_div_2exp} will require bit shifts,
but the choice is between an exponent in limbs which requires shifts there, or
one in bits which requires them almost everywhere else.

@item Use of @code{_mp_prec+1} Limbs
The extra limb on @code{_mp_d} (@code{_mp_prec+1} rather than just
@code{_mp_prec}) helps when an @code{mpf} routine might get a carry from its
operation.  @code{mpf_add} for instance will do an @code{mpn_add} of
@code{_mp_prec} limbs.  If there's no carry then that's the result, but if
there is a carry then it's stored in the extra limb of space and
@code{_mp_size} becomes @code{_mp_prec+1}.

Whenever @code{_mp_prec+1} limbs are held in a variable, the low limb is not
needed for the intended precision, only the @code{_mp_prec} high limbs.  But
zeroing it out or moving the rest down is unnecessary.  Subsequent routines
reading the value will simply take the high limbs they need, and this will be
@code{_mp_prec} if their target has that same precision.  This is no more than
a pointer adjustment, and must be checked anyway since the destination
precision can be different from the sources.

Copy functions like @code{mpf_set} will retain a full @code{_mp_prec+1} limbs
if available.  This ensures that a variable which has @code{_mp_size} equal to
@code{_mp_prec+1} will get its full exact value copied.  Strictly speaking
this is unnecessary since only @code{_mp_prec} limbs are needed for the
application's requested precision, but it's considered that an @code{mpf_set}
from one variable into another of the same precision ought to produce an exact
copy.

@item Application Precisions
@code{__GMPF_BITS_TO_PREC} converts an application requested precision to an
@code{_mp_prec}.  The value in bits is rounded up to a whole limb then an
extra limb is added since the most significant limb of @code{_mp_d} is only
non-zero and therefore might contain only one bit.

@code{__GMPF_PREC_TO_BITS} does the reverse conversion, and removes the extra
limb from @code{_mp_prec} before converting to bits.  The net effect of
reading back with @code{mpf_get_prec} is simply the precision rounded up to a
multiple of @code{mp_bits_per_limb}.

Note that the extra limb added here for the high only being non-zero is in
addition to the extra limb allocated to @code{_mp_d}.  For example with a
32-bit limb, an application request for 250 bits will be rounded up to 8
limbs, then an extra added for the high being only non-zero, giving an
@code{_mp_prec} of 9.  @code{_mp_d} then gets 10 limbs allocated.  Reading
back with @code{mpf_get_prec} will take @code{_mp_prec} subtract 1 limb and
multiply by 32, giving 256 bits.

Strictly speaking, the fact the high limb has at least one bit means that a
float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but
for the purposes of @code{mpf_t} it's considered simply to be 64 bits, a nice
multiple of the limb size.
@end table


@node Raw Output Internals, C++ Interface Internals, Float Internals, Internals
@section Raw Output Internals
@cindex Raw output internals

@noindent
@code{mpz_out_raw} uses the following format.

@tex
\global\newdimen\GMPboxwidth \GMPboxwidth=5em
\global\newdimen\GMPboxheight \GMPboxheight=3ex
\def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
\GMPdisplay{%
\vbox{%
  \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
  \vbox {%
    \hrule
    \hbox{%
      \vrule height 2.5ex depth 1.5ex
      \hbox to \GMPboxwidth {\hfil size\hfil}%
      \vrule
      \hbox to 3\GMPboxwidth {\hfil data bytes\hfil}%
      \vrule}
    \hrule}
}}
@end tex
@ifnottex
@example
+------+------------------------+
| size |       data bytes       |
+------+------------------------+
@end example
@end ifnottex

The size is 4 bytes written most significant byte first, being the number of
subsequent data bytes, or the twos complement negative of that when a negative
integer is represented.  The data bytes are the absolute value of the integer,
written most significant byte first.

The most significant data byte is always non-zero, so the output is the same
on all systems, irrespective of limb size.

In GMP 1, leading zero bytes were written to pad the data bytes to a multiple
of the limb size.  @code{mpz_inp_raw} will still accept this, for
compatibility.

The use of ``big endian'' for both the size and data fields is deliberate, it
makes the data easy to read in a hex dump of a file.  Unfortunately it also
means that the limb data must be reversed when reading or writing, so neither
a big endian nor little endian system can just read and write @code{_mp_d}.


@node C++ Interface Internals,  , Raw Output Internals, Internals
@section C++ Interface Internals
@cindex C++ interface internals

A system of expression templates is used to ensure something like @code{a=b+c}
turns into a simple call to @code{mpz_add} etc.  For @code{mpf_class}
the scheme also ensures the precision of the final
destination is used for any temporaries within a statement like
@code{f=w*x+y*z}.  These are important features which a naive implementation
cannot provide.

A simplified description of the scheme follows.  The true scheme is
complicated by the fact that expressions have different return types.  For
detailed information, refer to the source code.

To perform an operation, say, addition, we first define a ``function object''
evaluating it,

@example
struct __gmp_binary_plus
@{
  static void eval(mpf_t f, const mpf_t g, const mpf_t h)
  @{
    mpf_add(f, g, h);
  @}
@};
@end example

@noindent
And an ``additive expression'' object,

@example
__gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
operator+(const mpf_class &f, const mpf_class &g)
@{
  return __gmp_expr
    <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
@}
@end example

The seemingly redundant @code{__gmp_expr<__gmp_binary_expr<@dots{}>>} is used to
encapsulate any possible kind of expression into a single template type.  In
fact even @code{mpf_class} etc are @code{typedef} specializations of
@code{__gmp_expr}.

Next we define assignment of @code{__gmp_expr} to @code{mpf_class}.

@example
template <class T>
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
@{
  expr.eval(this->get_mpf_t(), this->precision());
  return *this;
@}

template <class Op>
void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
(mpf_t f, mp_bitcnt_t precision)
@{
  Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
@}
@end example

where @code{expr.val1} and @code{expr.val2} are references to the expression's
operands (here @code{expr} is the @code{__gmp_binary_expr} stored within the
@code{__gmp_expr}).

This way, the expression is actually evaluated only at the time of assignment,
when the required precision (that of @code{f}) is known.  Furthermore the
target @code{mpf_t} is now available, thus we can call @code{mpf_add} directly
with @code{f} as the output argument.

Compound expressions are handled by defining operators taking subexpressions
as their arguments, like this:

@example
template <class T, class U>
__gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
@{
  return __gmp_expr
    <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
    (expr1, expr2);
@}
@end example

And the corresponding specializations of @code{__gmp_expr::eval}:

@example
template <class T, class U, class Op>
void __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
(mpf_t f, mp_bitcnt_t precision)
@{
  // declare two temporaries
  mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
  Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
@}
@end example

The expression is thus recursively evaluated to any level of complexity and
all subexpressions are evaluated to the precision of @code{f}.


@node Contributors, References, Internals, Top
@comment  node-name,  next,  previous,  up
@appendix Contributors
@cindex Contributors

Torbj@"orn Granlund wrote the original GMP library and is still the main
developer.  Code not explicitly attributed to others, was contributed by
Torbj@"orn.  Several other individuals and organizations have contributed
GMP.  Here is a list in chronological order on first contribution:

Gunnar Sj@"odin and Hans Riesel helped with mathematical problems in early
versions of the library.

Richard Stallman helped with the interface design and revised the first
version of this manual.

Brian Beuning and Doug Lea helped with testing of early versions of the
library and made creative suggestions.

John Amanatides of York University in Canada contributed the function
@code{mpz_probab_prime_p}.

Paul Zimmermann wrote the REDC-based mpz_powm code, the Sch@"onhage-Strassen
FFT multiply code, and the Karatsuba square root code.  He also improved the
Toom3 code for GMP 4.2.  Paul sparked the development of GMP 2, with his
comparisons between bignum packages.  The ECMNET project Paul is organizing
was a driving force behind many of the optimizations in GMP 3.  Paul also
wrote the new GMP 4.3 nth root code (with Torbj@"orn).

Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul)
contributed now defunct versions of @code{mpz_gcd}, @code{mpz_divexact},
@code{mpn_gcd}, and @code{mpn_bdivmod}, partially supported by CNPq (Brazil)
grant 301314194-2.

Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure.
He has also made valuable suggestions and tested numerous intermediary
releases.

Joachim Hollman was involved in the design of the @code{mpf} interface, and in
the @code{mpz} design revisions for version 2.

Bennet Yee contributed the initial versions of @code{mpz_jacobi} and
@code{mpz_legendre}.

Andreas Schwab contributed the files @file{mpn/m68k/lshift.S} and
@file{mpn/m68k/rshift.S} (now in @file{.asm} form).

Robert Harley of Inria, France and David Seal of ARM, England, suggested clever
improvements for population count.  Robert also wrote highly optimized
Karatsuba and 3-way Toom multiplication functions for GMP 3, and contributed
the ARM assembly code.

Torsten Ekedahl of the Mathematical department of Stockholm University provided
significant inspiration during several phases of the GMP development.  His
mathematical expertise helped improve several algorithms.

Linus Nordberg wrote the new configure system based on autoconf and
implemented the new random functions.

Kevin Ryde worked on a large number of things: optimized x86 code, m4 asm
macros, parameter tuning, speed measuring, the configure system, function
inlining, divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas
number functions, printf and scanf functions, perl interface, demo expression
parser, the algorithms chapter in the manual, @file{gmpasm-mode.el}, and
various miscellaneous improvements elsewhere.

Kent Boortz made the Mac OS 9 port.

Steve Root helped write the optimized alpha 21264 assembly code.

Gerardo Ballabio wrote the @file{gmpxx.h} C++ class interface and the C++
@code{istream} input routines.

Jason Moxham rewrote @code{mpz_fac_ui}.

Pedro Gimeno implemented the Mersenne Twister and made other random number
improvements.

Niels M@"oller wrote the sub-quadratic GCD, extended GCD and jacobi code, the
quadratic Hensel division code, and (with Torbj@"orn) the new divide and
conquer division code for GMP 4.3.  Niels also helped implement the new Toom
multiply code for GMP 4.3 and implemented helper functions to simplify Toom
evaluations for GMP 5.0.  He wrote the original version of mpn_mulmod_bnm1, and
he is the main author of the mini-gmp package used for gmp bootstrapping.

Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply strategy,
and found the optimal strategies for evaluation and interpolation in Toom
multiplication.

Marco Bodrato helped implement the new Toom multiply code for GMP 4.3 and
implemented most of the new Toom multiply and squaring code for 5.0.
He is the main author of the current mpn_mulmod_bnm1, mpn_mullo_n, and
mpn_sqrlo.  Marco also wrote the functions mpn_invert and mpn_invertappr,
and improved the speed of integer root extraction.  He is the author of
mini-mpq, an additional layer to mini-gmp; of most of the combinatorial
functions and the BPSW primality testing implementation, for both the
main library and the mini-gmp package.

David Harvey suggested the internal function @code{mpn_bdiv_dbm1}, implementing
division relevant to Toom multiplication.  He also worked on fast assembly
sequences, in particular on a fast AMD64 @code{mpn_mul_basecase}. He wrote
the internal middle product functions @code{mpn_mulmid_basecase},
@code{mpn_toom42_mulmid}, @code{mpn_mulmid_n} and related helper routines.

Martin Boij wrote @code{mpn_perfect_power_p}.

Marc Glisse improved @file{gmpxx.h}: use fewer temporaries (faster),
specializations of @code{numeric_limits} and @code{common_type}, C++11
features (move constructors, explicit bool conversion, UDL), make the
conversion from @code{mpq_class} to @code{mpz_class} explicit, optimize
operations where one argument is a small compile-time constant, replace
some heap allocations by stack allocations.  He also fixed the eofbit
handling of C++ streams, and removed one division from @file{mpq/aors.c}.

David S Miller wrote assembly code for SPARC T3 and T4.

Mark Sofroniou cleaned up the types of mul_fft.c, letting it work for huge
operands.

Ulrich Weigand ported GMP to the powerpc64le ABI.

(This list is chronological, not ordered after significance.  If you have
contributed to GMP but are not listed above, please tell
@email{gmp-devel@@gmplib.org} about the omission!)

The development of floating point functions of GNU MP 2, were supported in part
by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial
System SOlving).

The development of GMP 2, 3, and 4.0 was supported in part by the IDA Center
for Computing Sciences.

The development of GMP 4.3, 5.0, and 5.1 was supported in part by the Swedish
Foundation for Strategic Research.

Thanks go to Hans Thorsen for donating an SGI system for the GMP test system
environment.

@node References, GNU Free Documentation License, Contributors, Top
@comment  node-name,  next,  previous,  up
@appendix References
@cindex References

@c  FIXME: In tex, the @uref's are unhyphenated, which is good for clarity,
@c  but being long words they upset paragraph formatting (the preceding line
@c  can get badly stretched).  Would like an conditional @* style line break
@c  if the uref is too long to fit on the last line of the paragraph, but it's
@c  not clear how to do that.  For now explicit @texlinebreak{}s are used on
@c  paragraphs that come out bad.

@section Books

@itemize @bullet
@item
Jonathan M. Borwein and Peter B. Borwein, ``Pi and the AGM: A Study in
Analytic Number Theory and Computational Complexity'', Wiley, 1998.

@item
Richard Crandall and Carl Pomerance, ``Prime Numbers: A Computational
Perspective'', 2nd edition, Springer-Verlag, 2005.
@texlinebreak{} @uref{https://www.math.dartmouth.edu/~carlp/}

@item
Henri Cohen, ``A Course in Computational Algebraic Number Theory'', Graduate
Texts in Mathematics number 138, Springer-Verlag, 1993.
@texlinebreak{} @uref{https://www.math.u-bordeaux.fr/~cohen/}

@item
Donald E. Knuth, ``The Art of Computer Programming'', volume 2,
``Seminumerical Algorithms'', 3rd edition, Addison-Wesley, 1998.
@texlinebreak{} @uref{https://www-cs-faculty.stanford.edu/~knuth/taocp.html}

@item
John D. Lipson, ``Elements of Algebra and Algebraic Computing'',
The Benjamin Cummings Publishing Company Inc, 1981.

@item
Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, ``Handbook of
Applied Cryptography'', @uref{http://www.cacr.math.uwaterloo.ca/hac/}

@item
Richard M. Stallman and the GCC Developer Community, ``Using the GNU Compiler
Collection'', Free Software Foundation, 2008, available online
@uref{https://gcc.gnu.org/onlinedocs/}, and in the GCC package
@uref{https://ftp.gnu.org/gnu/gcc/}
@end itemize

@section Papers

@itemize @bullet
@item
Yves Bertot, Nicolas Magaud and Paul Zimmermann, ``A Proof of GMP Square
Root'', Journal of Automated Reasoning, volume 29, 2002, pp.@: 225-252.  Also
available online as INRIA Research Report 4475, June 2002,
@uref{https://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf}

@item
Christoph Burnikel and Joachim Ziegler, ``Fast Recursive Division'',
Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
@texlinebreak{} @uref{https://www.mpi-inf.mpg.de/~ziegler/TechRep.ps.gz}

@item
Torbj@"orn Granlund and Peter L. Montgomery, ``Division by Invariant Integers
using Multiplication'', in Proceedings of the SIGPLAN PLDI'94 Conference, June
1994.  Also available @uref{https://gmplib.org/~tege/divcnst-pldi94.pdf}.

@item
Niels M@"oller and Torbj@"orn Granlund, ``Improved division by invariant
integers'', IEEE Transactions on Computers, 11 June 2010.
@uref{https://gmplib.org/~tege/division-paper.pdf}

@item
Torbj@"orn Granlund and Niels M@"oller, ``Division of integers large and
small'', to appear.

@item
Tudor Jebelean,
``An algorithm for exact division'',
Journal of Symbolic Computation,
volume 15, 1993, pp.@: 169-180.
Research report version available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz}

@item
Tudor Jebelean, ``Exact Division with Karatsuba Complexity - Extended
Abstract'', RISC-Linz technical report 96-31, @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz}

@item
Tudor Jebelean, ``Practical Integer Division with Karatsuba Complexity'',
ISSAC 97, pp.@: 339-341.  Technical report available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz}

@item
Tudor Jebelean, ``A Generalization of the Binary GCD Algorithm'', ISSAC 93,
pp.@: 111-116.  Technical report version available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz}

@item
Tudor Jebelean, ``A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD
of Long Integers'', Journal of Symbolic Computation, volume 19, 1995,
pp.@: 145-157.  Technical report version also available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz}

@item
Werner Krandick and Tudor Jebelean, ``Bidirectional Exact Integer Division'',
Journal of Symbolic Computation, volume 21, 1996, pp.@: 441-455.  Early
technical report version also available
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz}

@item
Makoto Matsumoto and Takuji Nishimura, ``Mersenne Twister: A 623-dimensionally
equidistributed uniform pseudorandom number generator'', ACM Transactions on
Modelling and Computer Simulation, volume 8, January 1998, pp.@: 3-30.
Available online @texlinebreak{}
@uref{http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.pdf}

@item
R. Moenck and A. Borodin, ``Fast Modular Transforms via Division'',
Proceedings of the 13th Annual IEEE Symposium on Switching and Automata
Theory, October 1972, pp.@: 90-96.  Reprinted as ``Fast Modular Transforms'',
Journal of Computer and System Sciences, volume 8, number 3, June 1974,
pp.@: 366-386.

@item
Niels M@"oller, ``On Sch@"onhage's algorithm and subquadratic integer GCD
  computation'', in Mathematics of Computation, volume 77, January 2008, pp.@:
  589-607, @uref{https://www.ams.org/journals/mcom/2008-77-261/S0025-5718-07-02017-0/home.html}

@item
Peter L. Montgomery, ``Modular Multiplication Without Trial Division'', in
Mathematics of Computation, volume 44, number 170, April 1985.

@item
Arnold Sch@"onhage and Volker Strassen, ``Schnelle Multiplikation grosser
Zahlen'', Computing 7, 1971, pp.@: 281-292.

@item
Kenneth Weber, ``The accelerated integer GCD algorithm'',
ACM Transactions on Mathematical Software,
volume 21, number 1, March 1995, pp.@: 111-122.

@item
Paul Zimmermann, ``Karatsuba Square Root'', INRIA Research Report 3805,
November 1999, @uref{https://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf}

@item
Paul Zimmermann, ``A Proof of GMP Fast Division and Square Root
Implementations'', @texlinebreak{}
@uref{https://homepages.loria.fr/PZimmermann/papers/proof-div-sqrt.ps.gz}

@item
Dan Zuras, ``On Squaring and Multiplying Large Integers'', ARITH-11: IEEE
Symposium on Computer Arithmetic, 1993, pp.@: 260 to 271.  Reprinted as ``More
on Multiplying and Squaring Large Integers'', IEEE Transactions on Computers,
volume 43, number 8, August 1994, pp.@: 899-908.

@item
Niels M@"oller, ``Efficient computation of the Jacobi symbol'', @texlinebreak{}
@uref{https://arxiv.org/abs/1907.07795}
@end itemize

@node GNU Free Documentation License, Concept Index, References, Top
@appendix GNU Free Documentation License
@cindex GNU Free Documentation License
@cindex Free Documentation License
@cindex Documentation license
@include fdl-1.3.texi


@node Concept Index, Function Index, GNU Free Documentation License, Top
@comment  node-name,  next,  previous,  up
@unnumbered Concept Index
@printindex cp

@node Function Index,  , Concept Index, Top
@comment  node-name,  next,  previous,  up
@unnumbered Function and Type Index
@printindex fn

@bye

@c Local variables:
@c fill-column: 78
@c compile-command: "make gmp.info"
@c End:
